#include <bits/stdc++.h>
#pragma comment(linker, "/stack:256000000")
#pragma gcc optimize("O3")
#pragma gcc target("sse4")
using namespace std;
mt19937 rng(2391);
mt19937 gnr(chrono::high_resolution_clock::now().time_since_epoch().count());
template <typename A>
istream& operator>>(istream& fin, vector<A>& v) {
  for (auto it = v.begin(); it != v.end(); ++it) fin >> *it;
  return fin;
}
template <typename A, typename B>
istream& operator>>(istream& fin, pair<A, B>& p) {
  fin >> p.first >> p.second;
  return fin;
}
template <typename A, typename B>
pair<A, B> operator+(const pair<A, B>& a, const pair<A, B>& b) {
  return make_pair(a.first + b.first, a.second + b.second);
}
template <typename A, typename B>
pair<A, B> operator+=(pair<A, B>& a, const pair<A, B>& b) {
  a.first += b.first;
  a.second += b.second;
  return a;
}
template <typename A, typename B>
pair<A, B> operator-(const pair<A, B>& a, const pair<A, B>& b) {
  return make_pair(a.first - b.first, a.second - b.second);
}
template <typename A, typename B>
pair<A, B> operator-(const pair<A, B>& a) {
  return make_pair(-a.first, -a.second);
}
template <typename A, typename B>
pair<A, B>& operator++(pair<A, B>& a) {
  ++a.first;
  ++a.second;
  return a;
}
template <typename A, typename B>
pair<A, B>& operator--(pair<A, B>& a) {
  --a.first;
  --a.second;
  return a;
}
template <typename A>
vector<A>& operator++(vector<A>& a) {
  for (auto it = a.begin(); it != a.end(); ++it) ++*it;
  return a;
}
template <typename A>
vector<A>& operator--(vector<A>& a) {
  for (auto it = a.begin(); it != a.end(); ++it) --*it;
  return a;
}
template <typename A, typename B>
pair<A, B> operator++(pair<A, B>& a, int) {
  pair<A, B> b = a;
  ++a;
  return b;
}
template <typename A, typename B>
pair<A, B> operator--(pair<A, B>& a, int) {
  pair<A, B> b = a;
  --a;
  return b;
}
template <typename A>
vector<A>& operator++(vector<A>& a, int) {
  vector<A> b = a;
  ++a;
  return b;
}
template <typename A>
vector<A>& operator--(vector<A>& a, int) {
  vector<A> b = a;
  --a;
  return b;
}
vector<vector<int>> adjlist_from_edgelist(const vector<pair<int, int>>& e,
                                          const int& n) {
  vector<vector<int>> g(n);
  for (auto it = e.begin(); it != e.end(); ++it) {
    g[it->first].push_back(it->second);
    g[it->second].push_back(it->first);
  }
  return g;
}
template <typename A, typename B>
pair<A, B> operator-=(pair<A, B>& a, const pair<A, B>& b) {
  a.first -= b.first;
  a.second -= b.second;
  return a;
}
template <typename A>
A operator*(const pair<A, A>& p, const pair<A, A>& q) {
  return p.first * q.first + p.second * q.second;
}
template <typename A>
pair<A, A> operator*(const pair<A, A>& p, const A& q) {
  return make_pair(p.first * q, p.second * q);
}
template <typename A>
A operator%(const pair<A, A>& p, const pair<A, A>& q) {
  return p.first * q.second - p.second * q.first;
}
template <typename A>
A sq_len(const pair<A, A>& p) {
  return p * p;
}
template <typename A>
A sq_dist(const pair<A, A>& p, const pair<A, A>& q) {
  return sq_len(p - q);
}
template <typename A>
double len(const pair<A, A>& p) {
  return sqrt(sq_len(p));
}
template <typename A>
double dist(const pair<A, A>& p, const pair<A, A>& q) {
  return len(p - q);
}
template <typename A>
bool is_rhombus(const pair<A, A>& a, const pair<A, A>& b, const pair<A, A>& c,
                const pair<A, A>& d) {
  A x = sq_dist(a, b);
  A y = sq_dist(b, c);
  A z = sq_dist(c, d);
  A t = sq_dist(d, a);
  return ((x == y) && (y == z) && (z == t));
}
template <typename A>
bool is_rectangle(const pair<A, A>& a, const pair<A, A>& b, const pair<A, A>& c,
                  const pair<A, A>& d) {
  pair<A, A> x = a - b, y = b - c, z = c - d, t = d - a;
  return ((x * y == 0) && (y * z == 0) && (z * t == 0) && (t * x == 0));
}
template <typename A>
bool are_parallel(const pair<A, A>& a, const pair<A, A>& b) {
  return (a % b == 0);
}
template <typename A>
bool are_orthogonal(const pair<A, A>& a, const pair<A, A>& b) {
  return (a * b == 0);
}
template <typename A>
bool are_codirected(const pair<A, A>& a, const pair<A, A>& b) {
  return (are_parallel(a, b) && (a * b >= 0));
}
template <typename A>
bool is_parallelogram(const pair<A, A>& a, const pair<A, A>& b,
                      const pair<A, A>& c, const pair<A, A>& d) {
  return ((a - b) == (d - c));
}
template <typename A>
bool is_trapezoid(const pair<A, A>& a, const pair<A, A>& b, const pair<A, A>& c,
                  const pair<A, A>& d) {
  pair<A, A> x = a - b, z = d - c;
  return are_codirected(x, z);
}
template <typename A>
bool is_convex_polygon(const pair<A, A>& a, const pair<A, A>& b,
                       const pair<A, A>& c, const pair<A, A>& d) {
  pair<A, A> x = a - b, y = b - c, z = c - d, t = d - a;
  A p = x % y, q = y % z, r = z % t, s = t % x;
  return (((p >= 0) && (q >= 0) && (r >= 0) && (s >= 0)) ||
          ((p <= 0) && (q <= 0) && (r <= 0) && (s <= 0)));
}
template <typename A>
bool nprcs(const pair<A, A>& a, const pair<A, A>& c) {
  return ((a % c) >= 0);
}
template <typename A>
bool nprcs(const pair<A, A>& a, const pair<A, A>& b, const pair<A, A>& c) {
  return nprcs(a - b, a - c);
}
template <typename A>
bool npocs(const pair<A, A>& a, const pair<A, A>& c) {
  return ((a % c) <= 0);
}
template <typename A>
bool npocs(const pair<A, A>& a, const pair<A, A>& b, const pair<A, A>& c) {
  return npocs(a - b, a - c);
}
template <typename A>
bool prcs(const pair<A, A>& a, const pair<A, A>& c) {
  return ((a % c) > 0);
}
template <typename A>
bool prcs(const pair<A, A>& a, const pair<A, A>& b, const pair<A, A>& c) {
  return prcs(a - b, a - c);
}
template <typename A>
bool pocs(const pair<A, A>& a, const pair<A, A>& c) {
  return ((a % c) < 0);
}
template <typename A>
bool pocs(const pair<A, A>& a, const pair<A, A>& b, const pair<A, A>& c) {
  return pocs(a - b, a - c);
}
template <typename A>
bool different_sides(const pair<A, A>& a, const pair<A, A>& b,
                     const pair<A, A>& c, const pair<A, A>& d) {
  pair<A, A> x = b - a;
  A p = x % (c - b), q = x % (d - b);
  return (((p >= 0) && (q <= 0)) || ((p <= 0) && (q >= 0)));
}
template <typename A>
bool sharply_different_sides(const pair<A, A>& a, const pair<A, A>& b,
                             const pair<A, A>& c, const pair<A, A>& d) {
  pair<A, A> x = b - a;
  A p = x % (c - b), q = x % (d - b);
  return (((p > 0) && (q < 0)) || ((p < 0) && (q > 0)));
}
template <typename A>
pair<A, A> rot_90(const pair<A, A>& p) {
  return make_pair(-p.second, p.first);
}
template <typename A>
pair<A, A> rot_90(const pair<A, A>& p, const pair<A, A>& c) {
  return c + rot_90(p - c);
}
template <typename A>
bool intersecting_segments(const pair<A, A>& a, const pair<A, A>& b,
                           const pair<A, A>& c, const pair<A, A>& d) {
  return different_sides(a, b, c, d) && different_sides(c, d, a, b);
}
template <typename A>
bool sharply_intersecting_segments(const pair<A, A>& a, const pair<A, A>& b,
                                   const pair<A, A>& c, const pair<A, A>& d) {
  return sharply_different_sides(a, b, c, d) &&
         sharply_different_sides(c, d, a, b);
}
template <typename A>
pair<pair<A, A>, A> line_with_normal_vector_through_point(
    const pair<A, A>& normal, const pair<A, A>& point) {
  return make_pair(normal, -(normal * point));
}
template <typename A>
pair<pair<A, A>, A> serper(const pair<A, A>& a, const pair<A, A>& b) {
  pair<A, A> p = b - a;
  return make_pair(p + p, -(p * (a + b)));
}
pair<double, double> ints(const pair<double, double>& p, const double& a,
                          const pair<double, double>& q, const double& b) {
  double D = p % q;
  double E =
      pair<double, double>(-a, p.second) % pair<double, double>(-b, q.second);
  double F =
      pair<double, double>(p.first, -a) % pair<double, double>(q.first, -b);
  return pair<double, double>(E / D, F / D);
}
pair<double, double> circumcenter(const pair<double, double>& x,
                                  const pair<double, double>& y,
                                  const pair<double, double>& z) {
  auto p1 = serper(x, y), p2 = serper(x, z);
  return ints(p1.first, p1.second, p2.first, p2.second);
}
template <typename A>
pair<pair<A, A>, A> l_th_2(const pair<A, A>& p, const pair<A, A>& q) {
  return make_pair(make_pair(q.second - p.second, p.first - q.first), -p % q);
}
template <typename A>
vector<pair<double, double>> circle_intersection(const pair<pair<A, A>, A>& a,
                                                 const pair<pair<A, A>, A>& b) {
  pair<A, A> c = b.first - a.first;
  A rr1 = a.second * a.second, rr2 = b.second * b.second, cc = c * c,
    rrrr1 = rr1 * rr1, rrrr2 = rr2 * rr2, cccc = cc * cc,
    D = 2 * (rr1 * rr2 + rr2 * cc + cc * rr1) - (rrrr1 + rrrr2 + cccc);
  if (D >= 0) {
    double E = (((double)(rr1 - rr2)) / cc + 1) / 2;
    pair<double, double> baza =
        pair<double, double>(a.first.first, a.first.second) +
        pair<double, double>(c.first, c.second) * E;
    if (D) {
      double lll = sqrt((double)(D)) / (2 * cc);
      pair<A, A> cr = rot_90(c);
      pair<double, double> pmm =
          pair<double, double>(cr.first, cr.second) * lll;
      return {baza + pmm, baza - pmm};
    } else
      return vector<pair<double, double>>(1, baza);
  } else
    return vector<pair<double, double>>();
}
template <typename A, typename B>
pair<A, B> sopr(const pair<A, B>& p) {
  return make_pair(p.first, -p.second);
}
template <typename A>
bool on_segment(const pair<A, A>& a, const pair<A, A>& b, const pair<A, A>& c) {
  return are_codirected(b - a, c - b);
}
template <typename A>
pair<pair<A, A>, A> perpendicular(const pair<A, A>& line,
                                  const pair<A, A>& point) {
  return line_with_normal_vector_through_point(rot_90(line), point);
}
pair<double, double> projection(const pair<pair<double, double>, double>& line,
                                const pair<double, double>& point) {
  pair<pair<double, double>, double> temp = perpendicular(line.first, point);
  return ints(line.first, line.second, temp.first, temp.second);
}
pair<double, double> height(const pair<double, double>& a,
                            const pair<double, double>& b,
                            const pair<double, double>& c) {
  return projection(l_th_2(a, c), b);
}
pair<double, double> unitvector(const pair<double, double>& a) {
  return a * (1 / len(a));
}
template <typename T>
vector<int> z_function(const vector<T>& s) {
  int n = ((int)(s.size()));
  vector<int> z(n);
  int l = 0, r = 1;
  for (int i = 1; i < n; ++i) {
    z[i] = max(0, min(r - i, z[i - l]));
    while (i + z[i] < n && (s[i + z[i]] == s[z[i]])) ++z[i];
    if (r < i + z[i]) {
      l = i;
      r = i + z[i];
    }
  }
  return z;
}
pair<int, int> euc(const int& m, const int& n) {
  if (n == 0) return pair<int, int>((m >= 0) ? 1 : -1, 0);
  int t = m / n;
  pair<int, int> ans1 = euc(n, m - t * n);
  return pair<int, int>(ans1.second, ans1.first - ans1.second * t);
}
int prod(const int& a, const int& b, const int& M) {
  return ((long long)(a)) * b % M;
}
int sum(const int& a, const int& b, const int& M) {
  int c = a + b;
  return c >= M ? c - M : c;
}
int raz(const int& a, const int& b, const int& M) {
  int c = a - b;
  return c < 0 ? c + M : c;
}
const int LITTLE_BORDER = 400;
struct factorizator {
  int N;
  vector<int> pr;
  vector<int> md;
  vector<int> pw;
  vector<int> without_md;
  factorizator(const int& n) : N(n), md(n), pw(n), without_md(n, 1) {
    for (int i = 2; i < N; ++i) md[i] = i;
    for (int i = 2; i < N; ++i) {
      if (md[i] == i) pr.push_back(i);
      bool worth = true;
      for (int j = 0; worth && (j < ((int)(pr.size()))) && (pr[j] <= md[i]);
           ++j) {
        long long temp = ((long long)(pr[j])) * i;
        if (temp < N)
          md[((int)temp)] = pr[j];
        else
          worth = false;
      }
    }
    for (int i = 2; i < N; ++i) {
      int t = md[i], s = i / t;
      if (md[s] == t) {
        pw[i] = 1 + pw[s];
        without_md[i] = without_md[s];
      } else {
        pw[i] = 1;
        without_md[i] = s;
      }
    }
  }
  void known_factorization(int n, vector<pair<unsigned long long, int>>& v) {
    while (n > 1) {
      v.emplace_back(md[n], pw[n]);
      n = without_md[n];
    }
  }
  vector<pair<unsigned long long, int>> pollardRho(const unsigned long long& n);
  vector<pair<unsigned long long, int>> factor(unsigned long long n);
  template <typename T>
  vector<pair<T, int>> merge(const vector<pair<T, int>>& p1,
                             const vector<pair<T, int>>& p2) {
    vector<pair<T, int>> ans;
    int m = ((int)(p1.size()));
    int n = ((int)(p2.size()));
    int i = 0;
    int j = 0;
    while ((i < m) || (j < n)) {
      if (i < m) {
        if (j < n) {
          if (p1[i].first < p2[j].first)
            ans.push_back(p1[i++]);
          else if (p1[i].first > p2[j].first)
            ans.push_back(p2[j++]);
          else {
            ans.emplace_back(p1[i].first, p1[i].second + p2[j].second);
            ++i;
            ++j;
          }
        } else
          ans.push_back(p1[i++]);
      } else
        ans.push_back(p2[j++]);
    }
    return ans;
  }
};
factorizator fac(LITTLE_BORDER);
unsigned long long experimental_prodll(const unsigned long long& a,
                                       const unsigned long long& b,
                                       const unsigned long long& M,
                                       const int& helper) {
  if (M == 0) return 0;
  unsigned long long q = ((long double)a) * b / M;
  unsigned long long a1 = a * b - q * M;
  unsigned long long a2 = a1;
  int rm = M % helper;
  int ra = a % helper;
  int rb = b % helper;
  int rq = q % helper;
  int r1 = raz(prod(ra, rb, helper), prod(rq, rm, helper), helper);
  int r2 = r1;
  while (true) {
    if (a1 % helper == r1 && a1 < M) return a1;
    a1 -= M;
    r1 = raz(r1, rm, helper);
    a2 += M;
    r2 = sum(r2, rm, helper);
    if (a2 % helper == r2 && a2 < M) return a2;
  }
}
inline int find_helper(const unsigned long long& M) {
  int i = 30;
  while (M % fac.pr[i] == 0) ++i;
  return fac.pr[i];
}
unsigned long long prodll(const unsigned long long& a,
                          const unsigned long long& b,
                          const unsigned long long& M) {
  if (M == 0) return 0;
  unsigned long long helper = find_helper(M);
  return experimental_prodll(a, b, M, helper);
}
unsigned long long sumll(const unsigned long long& a,
                         const unsigned long long& b,
                         const unsigned long long& M) {
  if (a < M - b)
    return a + b;
  else
    return a + b - M;
}
unsigned long long razll(const unsigned long long& a,
                         const unsigned long long& b,
                         const unsigned long long& M) {
  if (a >= b)
    return a - b;
  else
    return a + M - b;
}
template <typename T>
T diff(const T& a, const T& b) {
  return (a > b) ? (a - b) : (b - a);
}
template <typename A>
bool angdis_cmp(const pair<A, A>& a, const pair<A, A>& b) {
  A p;
  if (p = a % b)
    return (p > 0);
  else
    return sq_len(a) < sq_len(b);
}
template <typename T>
int find_min_idx(const vector<T>& v) {
  int ans = 0, n = ((int)(v.size()));
  for (int i = 1; i < n; ++i)
    if (v[i] < v[ans]) ans = i;
  return ans;
}
template <typename T>
vector<int> convex_hull(vector<pair<T, T>>& a) {
  int n = ((int)(a.size()));
  if (n) {
    int mt = find_min_idx(a);
    pair<T, T> d = a[mt];
    for (int i = 0; i < n; ++i) a[i] -= d;
    vector<int> idx(n);
    for (int i = 0; i < n; ++i) idx[i] = i;
    sort(idx.begin(), idx.end(),
         [&](const int& l, const int& r) { return angdis_cmp(a[l], a[r]); });
    vector<int> h(1, idx.front());
    for (auto it = idx.begin() + 1; it != idx.end(); ++it) {
      auto temp = a.begin() + *it;
      if (((int)(h.size())) >= 2) {
        if (are_parallel(a[h.back()], *temp)) h.pop_back();
        while ((((int)(h.size())) >= 3) &&
               npocs(a[h[((int)(h.size())) - 2]], a[h.back()], *temp))
          h.pop_back();
      }
      h.push_back(*it);
    }
    for (int i = 0; i < n; ++i) a[i] += d;
    return h;
  } else
    return vector<int>();
}
pair<int, int> cool_gcd(const int& a, const int& b) {
  if (b) {
    int c = a / b;
    pair<int, int> ans1 = cool_gcd(b, a - b * c);
    return pair<long long, long long>(ans1.second,
                                      ans1.first - ans1.second * c);
  } else
    return pair<int, int>(1, 0);
}
pair<long long, long long> cool_gcdll(const long long& a, const long long& b) {
  if (b) {
    long long c = a / b;
    pair<long long, long long> ans1 = cool_gcdll(b, a - b * c);
    return pair<long long, long long>(ans1.second,
                                      ans1.first - ans1.second * c);
  } else
    return pair<long long, long long>(1ll, 0ll);
}
template <typename T>
T gcd(T a, T b) {
  while (b) {
    a %= b;
    swap(a, b);
  }
  return a;
}
long long pr_p(const long long& a, const long long& b, const long long& p) {
  if (b < 1000000) return (a * b) % p;
  long long temp = pr_p(a, b >> 1ll, p);
  temp = sumll(temp, temp, p);
  if (b & 1ll)
    return sumll(temp, a, p);
  else
    return temp;
}
long long po_p(const long long& a, const long long& b, const long long& p) {
  if (b < 2) {
    if (b == 0)
      return 1;
    else
      return a;
  }
  long long temp = po_p(a, b >> 1ll, p);
  temp = pr_p(temp, temp, p);
  if (b & 1ll)
    return pr_p(temp, a, p);
  else
    return temp;
}
int pow_mod(const int& a, const int& b, const int& p) {
  if (b < 2) {
    if (b == 0)
      return 1;
    else
      return a;
  }
  int temp = pow_mod(a, b >> 1, p);
  temp = prod(temp, temp, p);
  if (b & 1)
    return prod(temp, a, p);
  else
    return temp;
}
long long pow_modll(const long long& a, const long long& b,
                    const long long& p) {
  if (b < 2) {
    if (b == 0)
      return 1;
    else
      return a;
  }
  long long temp = pow_modll(a, b >> 1ll, p);
  temp = prodll(temp, temp, p);
  if (b & 1ll)
    return prodll(temp, a, p);
  else
    return temp;
}
int inverse(int a, int n) {
  int c = cool_gcd(a, n).first;
  if (c < 0) c += n;
  return c;
}
long long inversell(long long a, long long n) {
  long long c = cool_gcdll(a, n).first;
  if (c < 0) c += n;
  return c;
}
template <typename A>
pair<int, int> equal_elements(const vector<A>& u, const vector<A>& v) {
  pair<int, int> ans(INT_MAX, INT_MAX);
  int m = ((int)(u.size())), n = ((int)(v.size()));
  vector<int> id_u(m);
  for (int i = 1; i < m; ++i) id_u[i] = i;
  vector<int> id_v(n);
  for (int i = 1; i < n; ++i) id_v[i] = i;
  sort(id_u.begin(), id_u.end(),
       [&](const int& x, const int& y) { return u[x] < u[y]; });
  sort(id_v.begin(), id_v.end(),
       [&](const int& x, const int& y) { return v[x] < v[y]; });
  int i = 0;
  int j = 0;
  while ((i < m) && (j < n)) {
    if (u[id_u[i]] < v[id_v[j]])
      ++i;
    else if (v[id_v[j]] < u[id_u[i]])
      ++j;
    else {
      ans = min(ans, pair<int, int>(id_v[j], id_u[i]));
      ++j;
    }
  }
  if (ans.first == INT_MAX)
    return pair<int, int>(-1, -1);
  else
    return pair<int, int>(ans.second, ans.first);
}
long long discr_log(long long a, long long b, long long n) {
  int k = ((int)(sqrt((long double)n)));
  long long a1 = inversell(a, n);
  int l = k + 20;
  long long a2 = po_p(a1, k, n);
  vector<long long> seq1(k), seq2(l);
  seq1.front() = 1;
  for (int i = 1; i < k; ++i) seq1[i] = pr_p(seq1[i - 1], a, n);
  seq2.front() = b;
  for (int i = 1; i < l; ++i) seq2[i] = pr_p(seq2[i - 1], a2, n);
  pair<long long, long long> e = equal_elements(seq1, seq2);
  if (e.first == -1)
    return -1;
  else
    return e.first + e.second * k;
}
long long common_discr_log(long long a, long long b, long long n) {
  const int C = 70;
  a %= n;
  b %= n;
  if (gcd(n, a) != 1) {
    for (int i = 0; i < C; ++i) {
      if (po_p(a, i, n) == b) {
        return i;
      }
    }
    long long multp = po_p(a, C, n);
    long long g = gcd(multp, n);
    if (b % g) {
      return -1;
    } else {
      b /= g;
      n /= g;
      multp /= g;
      long long pop_back = inversell(multp, n);
      b = pr_p(b, pop_back, n);
      long long ans = discr_log(a, b, n);
      if (ans == -1)
        return -1;
      else
        return ans + C;
    }
  } else
    return discr_log(a, b, n);
}
const unsigned long long minimum_composites[] = {1ull,
                                                 2046ull,
                                                 1373652ull,
                                                 25326000ull,
                                                 3215031750ull,
                                                 2152302898746ull,
                                                 3474749660382ull,
                                                 341550071728320ull,
                                                 341550071728320ull,
                                                 3825123056546413050ull,
                                                 3825123056546413050ull,
                                                 18446744073709551615ull};
const int little_primes[] = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37};
bool miller_rabin_check(const unsigned long long& n,
                        const unsigned long long& prime,
                        const unsigned long long& odd,
                        const unsigned long long& pot,
                        const unsigned long long& nmo) {
  long long x = pow_modll(prime, odd, n);
  if ((x == nmo) || (x == 1)) return true;
  for (int i = 1; i < pot; ++i) {
    x = prodll(x, x, n);
    if (x == nmo) return true;
    if (x == 1) return false;
  }
  return false;
}
bool is_prime(const unsigned long long& n) {
  if (n <= 2) return (n == 2);
  unsigned long long odd, pot, nmo;
  nmo = n - 1;
  odd = nmo;
  pot = 0;
  while ((odd & 1ll) == 0) {
    odd >>= 1ll;
    ++pot;
  }
  int i = 0;
  while (n > minimum_composites[i]) {
    if (miller_rabin_check(n, little_primes[i], odd, pot, nmo) == false)
      return false;
    ++i;
  }
  return true;
}
vector<pair<unsigned long long, int>> factorizator::pollardRho(
    const unsigned long long& n) {
  if (is_prime(n))
    return vector<pair<unsigned long long, int>>(
        1, pair<unsigned long long, int>(n, 1));
  int step = 1;
  while (true) {
    step <<= 1;
    unsigned long long a = rng();
    unsigned long long b = a;
    unsigned long long d = 1;
    for (int j = 0; j < step; ++j) {
      a = sumll(1, prodll(a, a, n), n);
      b = sumll(1, prodll(b, b, n), n);
      b = sumll(1, prodll(b, b, n), n);
      d = gcd(diff(a, b), n);
      if (d > 1) {
        break;
      }
    }
    if (1 < d && d < n) return merge(pollardRho(d), pollardRho(n / d));
  }
  return vector<pair<unsigned long long, int>>();
}
vector<pair<unsigned long long, int>> factorizator::factor(
    unsigned long long n) {
  vector<pair<unsigned long long, int>> ans;
  for (int i = 0; n >= N && i < ((int)(pr.size())) && pr[i] < LITTLE_BORDER;
       ++i)
    while (n % pr[i] == 0) {
      if (((int)(ans.size())) && ans.back().first == pr[i])
        ++ans.back().second;
      else
        ans.emplace_back(pr[i], 1);
      n /= pr[i];
    }
  if (n < N) {
    known_factorization(n, ans);
    return ans;
  } else
    return merge(ans, pollardRho(n));
}
template <typename T>
T phi(const vector<pair<T, int>>& v) {
  T ans = 1;
  for (int i = 0; i < ((int)(v.size())); ++i) {
    ans *= v[i].first - 1;
    for (int j = 1; j < v[i].second; ++j) ans *= v[i].first;
  }
  return ans;
}
unsigned long long phi(const unsigned long long& n) {
  return phi(fac.factor(n));
}
bool check_primitive_root(const int& ph, const vector<int>& to_check,
                          const int& r, const int& n) {
  for (int i = 0; i < ((int)(to_check.size())); ++i)
    if (pow_mod(r, to_check[i], n) == 1) return false;
  return (pow_mod(r, ph, n) == 1);
}
int primitive_root(const int& n) {
  if (n < 3) return n - 1;
  int p = phi(n);
  vector<pair<unsigned long long, int>> f = fac.factor(p);
  vector<int> to_check(((int)(f.size())));
  for (int i = 0; i < ((int)(f.size())); ++i) to_check[i] = p / f[i].first;
  for (int i = 2; i < n; ++i)
    if (check_primitive_root(p, to_check, i, n)) return i;
  return -1;
}
int unite_mod(const int& a, const int& p, const int& b, const int& q) {
  pair<int, int> c = cool_gcd(p, q);
  int pr = p * q;
  int ans = ((a * c.second * q + b * c.first * p) % pr + pr) % pr;
  return ans;
}
long long unite_modll(const long long& a, const long long& p,
                      const long long& b, const long long& q) {
  pair<long long, long long> c = cool_gcdll(p, q);
  long long pr = p * q;
  long long ans = ((a * c.second * q + b * c.first * p) % pr + pr) % pr;
  return ans;
}
pair<int, int> power_v(int n, const int& p) {
  int ans = 0;
  while (n % p == 0) {
    n /= p;
    ++ans;
  }
  return pair<int, int>(ans, n);
}
int square_root_prime_modulo(int c, int n, const int& pr, const int& k) {
  c %= n;
  if (c) {
    pair<int, int> kek = power_v(c, pr);
    int l = kek.first;
    if (l & 1) return -1;
    if (l > 0) {
      int pwl = 1;
      for (int i = 0; i < l; ++i) pwl *= pr;
      n /= pwl;
      c /= pwl;
      int ans1 = square_root_prime_modulo(c, n, pr, k - l);
      if (ans1 == -1) return -1;
      for (int i = 0; i < (l >> 1); ++i) ans1 *= pr;
      return ans1;
    } else {
      int primitive;
      if (n & 1)
        primitive = primitive_root(n);
      else
        primitive = 5;
      int u = ((int)discr_log(primitive, c, n));
      if (u == -1) return -1;
      if (u & 1) return -1;
      return pow_mod(primitive, u >> 1, n);
    }
  } else
    return 0;
}
int square_root_modulo(const int& c, const int& n) {
  vector<pair<unsigned long long, int>> f = fac.factor(n);
  int a = 0, p = 1;
  for (int i = 0; i < ((int)(f.size())); ++i) {
    int q = 1;
    for (int j = 0; j < f[i].second; ++j) q *= f[i].first;
    int b = square_root_prime_modulo(c, q, f[i].first, f[i].second);
    if (b == -1) return -1;
    a = unite_mod(a, p, b, q);
    p *= q;
  }
  return a;
}
namespace BGF {
inline int cntd(unsigned x) {
  int ans = 0;
  while (x) {
    if (x & 1) ++ans;
    x >>= 1;
  }
  return ans;
}
inline long long cntdll(unsigned long long x) {
  long long ans = 0;
  while (x) {
    if (x & 1) ++ans;
    x >>= 1;
  }
  return ans;
}
inline int clzd(unsigned x) {
  int ans = 32;
  while (x) {
    --ans;
    x >>= 1;
  }
  return ans;
}
inline long long clzdll(unsigned long long x) {
  long long ans = 64;
  while (x) {
    --ans;
    x >>= 1;
  }
  return ans;
}
inline int ctzd(unsigned x) {
  if (x == 0) return 32;
  int ans = 0;
  while ((x & 1) == 0) {
    ++ans;
    x >>= 1;
  }
  return ans;
}
inline long long ctzdll(unsigned long long x) {
  if (x == 0) return 64;
  long long ans = 0;
  while ((x & 1) == 0) {
    ++ans;
    x >>= 1;
  }
  return ans;
}
}  // namespace BGF
namespace BGF {
const unsigned BHC = 65535u;
const unsigned long long BHCL = 65535ull;
const unsigned BHN = 65536u;
const unsigned BHL = 16u;
const unsigned LH = 4294901760u;
const unsigned RH = 65535u;
const unsigned long long BHLL1 = 16ull;
const unsigned long long BHLL2 = 32ull;
const unsigned long long BHLL3 = 48ull;
const unsigned long long LHL = 18446744069414584320ull;
const unsigned long long RHL = 4294967295ull;
int CNT[BHN];
int CLZ[BHN];
int CTZ[BHN];
inline void initialize() {
  CNT[0] = 0;
  CLZ[0] = BHL + BHL;
  CTZ[0] = BHL;
  for (unsigned i = 1; i < BHN; ++i) {
    CNT[i] = cntd(i);
    CLZ[i] = clzd(i);
    CTZ[i] = ctzd(i);
  }
}
inline int cntc(const unsigned& x) {
  return CNT[x & BHC] + CNT[(x >> BHL) & BHC];
}
inline long long cntcll(const unsigned long long& x) {
  return CNT[x & BHCL] + CNT[(x >> BHLL1) & BHCL] + CNT[(x >> BHLL2) & BHCL] +
         CNT[(x >> BHLL3) & BHCL];
}
inline int clzc(const unsigned& x) {
  if (x & LH)
    return CLZ[x >> BHL] - BHL;
  else
    return CLZ[x];
}
inline long long clzcll(const unsigned long long& x) {
  if (x & LHL)
    return clzc(x >> BHLL2);
  else
    return clzc(x) + BHLL2;
}
inline int ctzc(const unsigned& x) {
  if (x & RH)
    return CTZ[x & RH];
  else
    return CTZ[x >> BHL] + BHL;
}
inline long long ctzcll(const unsigned long long& x) {
  if (x & RHL)
    return ctzc(x);
  else
    return ctzc(x >> BHLL2) + BHLL2;
}
}  // namespace BGF
const string INPUT_FILE = "input.txt";
const string OUTPUT_FILE = "output.txt";
void relax(long long& a, const long long& b) { a = max(a, b); }
int another_main() {
  int n, k;
  cin >> n >> k;
  vector<int> a(k);
  cin >> a;
  --a;
  int N = 1 << n;
  vector<bool> v(N);
  for (int i : a) v[i] = true;
  N /= 2;
  vector<vector<long long>> dp(N, vector<long long>(4, -1e9));
  for (int i = 0; i < N; ++i) {
    int x = v[2 * i];
    int y = v[2 * i + 1];
    int dob = (x || y);
    dp[i][2 * x + y] = dob;
    dp[i][2 * y + x] = dob;
  }
  while (N > 1) {
    N /= 2;
    vector<vector<long long>> dq(N, vector<long long>(4, -1e9));
    for (int i = 0; i < N; ++i)
      for (int j = 0; j < 4; ++j)
        for (int l = 0; l < 4; ++l) {
          int base11 = (j & l & 1);
          int base1 = (((j & 1) | (l & 1)) != 0);
          int base2 = (((j & 2) | (l & 2)) != 0);
          long long base3 = dp[2 * i][j] + dp[2 * i + 1][l];
          long long base = base1 + base2 + base3;
          int p = -3;
          if (base2 || base11)
            p = 2 | base1;
          else if (base2 == 0 && base1 == 0)
            p = 0;
          if (p != -3) {
            relax(dq[i][p], base + (p != 0));
            if (dq[i][p] == 5) {
              i = i;
            }
          } else {
            relax(dq[i][1], base);
            relax(dq[i][2], base + 1);
          }
        }
    dp = dq;
  }
  long long base0 = dp[0][0];
  long long base1 = dp[0][1] + 1;
  long long base2 = dp[0][2] + 1;
  long long base3 = dp[0][3] + 1;
  cout << max(max(base0, base1), max(base2, base3)) << '\n';
  return 0;
}
int main() {
  ios_base::sync_with_stdio(0);
  cin.tie(0);
  cout.tie(0);
  cerr.tie(0);
  BGF::initialize();
  int AMR = another_main();
  return AMR;
}

### Prompt

Construct a cpp code solution to the problem outlined:

The biggest event of the year – Cota 2 world championship "The Innernational" is right around the corner. 2^n teams will compete in a double-elimination format (please, carefully read problem statement even if you know what is it) to identify the champion. 

Teams are numbered from 1 to 2^n and will play games one-on-one. All teams start in the upper bracket.

All upper bracket matches will be held played between teams that haven't lost any games yet. Teams are split into games by team numbers. Game winner advances in the next round of upper bracket, losers drop into the lower bracket.

Lower bracket starts with 2^{n-1} teams that lost the first upper bracket game. Each lower bracket round consists of two games. In the first game of a round 2^k teams play a game with each other (teams are split into games by team numbers). 2^{k-1} loosing teams are eliminated from the championship, 2^{k-1} winning teams are playing 2^{k-1} teams that got eliminated in this round of upper bracket (again, teams are split into games by team numbers). As a result of each round both upper and lower bracket have 2^{k-1} teams remaining. See example notes for better understanding.

Single remaining team of upper bracket plays with single remaining team of lower bracket in grand-finals to identify championship winner.

You are a fan of teams with numbers a_1, a_2, ..., a_k. You want the championship to have as many games with your favourite teams as possible. Luckily, you can affect results of every championship game the way you want. What's maximal possible number of championship games that include teams you're fan of?

Input

First input line has two integers n, k — 2^n teams are competing in the championship. You are a fan of k teams (2 ≤ n ≤ 17; 0 ≤ k ≤ 2^n).

Second input line has k distinct integers a_1, …, a_k — numbers of teams you're a fan of (1 ≤ a_i ≤ 2^n).

Output

Output single integer — maximal possible number of championship games that include teams you're fan of.

Examples

Input


3 1
6


Output


6


Input


3 3
1 7 8


Output


11


Input


3 4
1 3 5 7


Output


14

Note

On the image, each game of the championship is denoted with an English letter (a to n). Winner of game i is denoted as Wi, loser is denoted as Li. Teams you're a fan of are highlighted with red background.

In the first example, team 6 will play in 6 games if it looses the first upper bracket game (game c) and wins all lower bracket games (games h, j, l, m). 

<image>

In the second example, teams 7 and 8 have to play with each other in the first game of upper bracket (game d). Team 8 can win all remaining games in upper bracket, when teams 1 and 7 will compete in the lower bracket. 

<image>

In the third example, your favourite teams can play in all games of the championship. 

<image>

### Response

```cpp
#include <bits/stdc++.h>
#pragma comment(linker, "/stack:256000000")
#pragma gcc optimize("O3")
#pragma gcc target("sse4")
using namespace std;
mt19937 rng(2391);
mt19937 gnr(chrono::high_resolution_clock::now().time_since_epoch().count());
template <typename A>
istream& operator>>(istream& fin, vector<A>& v) {
  for (auto it = v.begin(); it != v.end(); ++it) fin >> *it;
  return fin;
}
template <typename A, typename B>
istream& operator>>(istream& fin, pair<A, B>& p) {
  fin >> p.first >> p.second;
  return fin;
}
template <typename A, typename B>
pair<A, B> operator+(const pair<A, B>& a, const pair<A, B>& b) {
  return make_pair(a.first + b.first, a.second + b.second);
}
template <typename A, typename B>
pair<A, B> operator+=(pair<A, B>& a, const pair<A, B>& b) {
  a.first += b.first;
  a.second += b.second;
  return a;
}
template <typename A, typename B>
pair<A, B> operator-(const pair<A, B>& a, const pair<A, B>& b) {
  return make_pair(a.first - b.first, a.second - b.second);
}
template <typename A, typename B>
pair<A, B> operator-(const pair<A, B>& a) {
  return make_pair(-a.first, -a.second);
}
template <typename A, typename B>
pair<A, B>& operator++(pair<A, B>& a) {
  ++a.first;
  ++a.second;
  return a;
}
template <typename A, typename B>
pair<A, B>& operator--(pair<A, B>& a) {
  --a.first;
  --a.second;
  return a;
}
template <typename A>
vector<A>& operator++(vector<A>& a) {
  for (auto it = a.begin(); it != a.end(); ++it) ++*it;
  return a;
}
template <typename A>
vector<A>& operator--(vector<A>& a) {
  for (auto it = a.begin(); it != a.end(); ++it) --*it;
  return a;
}
template <typename A, typename B>
pair<A, B> operator++(pair<A, B>& a, int) {
  pair<A, B> b = a;
  ++a;
  return b;
}
template <typename A, typename B>
pair<A, B> operator--(pair<A, B>& a, int) {
  pair<A, B> b = a;
  --a;
  return b;
}
template <typename A>
vector<A>& operator++(vector<A>& a, int) {
  vector<A> b = a;
  ++a;
  return b;
}
template <typename A>
vector<A>& operator--(vector<A>& a, int) {
  vector<A> b = a;
  --a;
  return b;
}
vector<vector<int>> adjlist_from_edgelist(const vector<pair<int, int>>& e,
                                          const int& n) {
  vector<vector<int>> g(n);
  for (auto it = e.begin(); it != e.end(); ++it) {
    g[it->first].push_back(it->second);
    g[it->second].push_back(it->first);
  }
  return g;
}
template <typename A, typename B>
pair<A, B> operator-=(pair<A, B>& a, const pair<A, B>& b) {
  a.first -= b.first;
  a.second -= b.second;
  return a;
}
template <typename A>
A operator*(const pair<A, A>& p, const pair<A, A>& q) {
  return p.first * q.first + p.second * q.second;
}
template <typename A>
pair<A, A> operator*(const pair<A, A>& p, const A& q) {
  return make_pair(p.first * q, p.second * q);
}
template <typename A>
A operator%(const pair<A, A>& p, const pair<A, A>& q) {
  return p.first * q.second - p.second * q.first;
}
template <typename A>
A sq_len(const pair<A, A>& p) {
  return p * p;
}
template <typename A>
A sq_dist(const pair<A, A>& p, const pair<A, A>& q) {
  return sq_len(p - q);
}
template <typename A>
double len(const pair<A, A>& p) {
  return sqrt(sq_len(p));
}
template <typename A>
double dist(const pair<A, A>& p, const pair<A, A>& q) {
  return len(p - q);
}
template <typename A>
bool is_rhombus(const pair<A, A>& a, const pair<A, A>& b, const pair<A, A>& c,
                const pair<A, A>& d) {
  A x = sq_dist(a, b);
  A y = sq_dist(b, c);
  A z = sq_dist(c, d);
  A t = sq_dist(d, a);
  return ((x == y) && (y == z) && (z == t));
}
template <typename A>
bool is_rectangle(const pair<A, A>& a, const pair<A, A>& b, const pair<A, A>& c,
                  const pair<A, A>& d) {
  pair<A, A> x = a - b, y = b - c, z = c - d, t = d - a;
  return ((x * y == 0) && (y * z == 0) && (z * t == 0) && (t * x == 0));
}
template <typename A>
bool are_parallel(const pair<A, A>& a, const pair<A, A>& b) {
  return (a % b == 0);
}
template <typename A>
bool are_orthogonal(const pair<A, A>& a, const pair<A, A>& b) {
  return (a * b == 0);
}
template <typename A>
bool are_codirected(const pair<A, A>& a, const pair<A, A>& b) {
  return (are_parallel(a, b) && (a * b >= 0));
}
template <typename A>
bool is_parallelogram(const pair<A, A>& a, const pair<A, A>& b,
                      const pair<A, A>& c, const pair<A, A>& d) {
  return ((a - b) == (d - c));
}
template <typename A>
bool is_trapezoid(const pair<A, A>& a, const pair<A, A>& b, const pair<A, A>& c,
                  const pair<A, A>& d) {
  pair<A, A> x = a - b, z = d - c;
  return are_codirected(x, z);
}
template <typename A>
bool is_convex_polygon(const pair<A, A>& a, const pair<A, A>& b,
                       const pair<A, A>& c, const pair<A, A>& d) {
  pair<A, A> x = a - b, y = b - c, z = c - d, t = d - a;
  A p = x % y, q = y % z, r = z % t, s = t % x;
  return (((p >= 0) && (q >= 0) && (r >= 0) && (s >= 0)) ||
          ((p <= 0) && (q <= 0) && (r <= 0) && (s <= 0)));
}
template <typename A>
bool nprcs(const pair<A, A>& a, const pair<A, A>& c) {
  return ((a % c) >= 0);
}
template <typename A>
bool nprcs(const pair<A, A>& a, const pair<A, A>& b, const pair<A, A>& c) {
  return nprcs(a - b, a - c);
}
template <typename A>
bool npocs(const pair<A, A>& a, const pair<A, A>& c) {
  return ((a % c) <= 0);
}
template <typename A>
bool npocs(const pair<A, A>& a, const pair<A, A>& b, const pair<A, A>& c) {
  return npocs(a - b, a - c);
}
template <typename A>
bool prcs(const pair<A, A>& a, const pair<A, A>& c) {
  return ((a % c) > 0);
}
template <typename A>
bool prcs(const pair<A, A>& a, const pair<A, A>& b, const pair<A, A>& c) {
  return prcs(a - b, a - c);
}
template <typename A>
bool pocs(const pair<A, A>& a, const pair<A, A>& c) {
  return ((a % c) < 0);
}
template <typename A>
bool pocs(const pair<A, A>& a, const pair<A, A>& b, const pair<A, A>& c) {
  return pocs(a - b, a - c);
}
template <typename A>
bool different_sides(const pair<A, A>& a, const pair<A, A>& b,
                     const pair<A, A>& c, const pair<A, A>& d) {
  pair<A, A> x = b - a;
  A p = x % (c - b), q = x % (d - b);
  return (((p >= 0) && (q <= 0)) || ((p <= 0) && (q >= 0)));
}
template <typename A>
bool sharply_different_sides(const pair<A, A>& a, const pair<A, A>& b,
                             const pair<A, A>& c, const pair<A, A>& d) {
  pair<A, A> x = b - a;
  A p = x % (c - b), q = x % (d - b);
  return (((p > 0) && (q < 0)) || ((p < 0) && (q > 0)));
}
template <typename A>
pair<A, A> rot_90(const pair<A, A>& p) {
  return make_pair(-p.second, p.first);
}
template <typename A>
pair<A, A> rot_90(const pair<A, A>& p, const pair<A, A>& c) {
  return c + rot_90(p - c);
}
template <typename A>
bool intersecting_segments(const pair<A, A>& a, const pair<A, A>& b,
                           const pair<A, A>& c, const pair<A, A>& d) {
  return different_sides(a, b, c, d) && different_sides(c, d, a, b);
}
template <typename A>
bool sharply_intersecting_segments(const pair<A, A>& a, const pair<A, A>& b,
                                   const pair<A, A>& c, const pair<A, A>& d) {
  return sharply_different_sides(a, b, c, d) &&
         sharply_different_sides(c, d, a, b);
}
template <typename A>
pair<pair<A, A>, A> line_with_normal_vector_through_point(
    const pair<A, A>& normal, const pair<A, A>& point) {
  return make_pair(normal, -(normal * point));
}
template <typename A>
pair<pair<A, A>, A> serper(const pair<A, A>& a, const pair<A, A>& b) {
  pair<A, A> p = b - a;
  return make_pair(p + p, -(p * (a + b)));
}
pair<double, double> ints(const pair<double, double>& p, const double& a,
                          const pair<double, double>& q, const double& b) {
  double D = p % q;
  double E =
      pair<double, double>(-a, p.second) % pair<double, double>(-b, q.second);
  double F =
      pair<double, double>(p.first, -a) % pair<double, double>(q.first, -b);
  return pair<double, double>(E / D, F / D);
}
pair<double, double> circumcenter(const pair<double, double>& x,
                                  const pair<double, double>& y,
                                  const pair<double, double>& z) {
  auto p1 = serper(x, y), p2 = serper(x, z);
  return ints(p1.first, p1.second, p2.first, p2.second);
}
template <typename A>
pair<pair<A, A>, A> l_th_2(const pair<A, A>& p, const pair<A, A>& q) {
  return make_pair(make_pair(q.second - p.second, p.first - q.first), -p % q);
}
template <typename A>
vector<pair<double, double>> circle_intersection(const pair<pair<A, A>, A>& a,
                                                 const pair<pair<A, A>, A>& b) {
  pair<A, A> c = b.first - a.first;
  A rr1 = a.second * a.second, rr2 = b.second * b.second, cc = c * c,
    rrrr1 = rr1 * rr1, rrrr2 = rr2 * rr2, cccc = cc * cc,
    D = 2 * (rr1 * rr2 + rr2 * cc + cc * rr1) - (rrrr1 + rrrr2 + cccc);
  if (D >= 0) {
    double E = (((double)(rr1 - rr2)) / cc + 1) / 2;
    pair<double, double> baza =
        pair<double, double>(a.first.first, a.first.second) +
        pair<double, double>(c.first, c.second) * E;
    if (D) {
      double lll = sqrt((double)(D)) / (2 * cc);
      pair<A, A> cr = rot_90(c);
      pair<double, double> pmm =
          pair<double, double>(cr.first, cr.second) * lll;
      return {baza + pmm, baza - pmm};
    } else
      return vector<pair<double, double>>(1, baza);
  } else
    return vector<pair<double, double>>();
}
template <typename A, typename B>
pair<A, B> sopr(const pair<A, B>& p) {
  return make_pair(p.first, -p.second);
}
template <typename A>
bool on_segment(const pair<A, A>& a, const pair<A, A>& b, const pair<A, A>& c) {
  return are_codirected(b - a, c - b);
}
template <typename A>
pair<pair<A, A>, A> perpendicular(const pair<A, A>& line,
                                  const pair<A, A>& point) {
  return line_with_normal_vector_through_point(rot_90(line), point);
}
pair<double, double> projection(const pair<pair<double, double>, double>& line,
                                const pair<double, double>& point) {
  pair<pair<double, double>, double> temp = perpendicular(line.first, point);
  return ints(line.first, line.second, temp.first, temp.second);
}
pair<double, double> height(const pair<double, double>& a,
                            const pair<double, double>& b,
                            const pair<double, double>& c) {
  return projection(l_th_2(a, c), b);
}
pair<double, double> unitvector(const pair<double, double>& a) {
  return a * (1 / len(a));
}
template <typename T>
vector<int> z_function(const vector<T>& s) {
  int n = ((int)(s.size()));
  vector<int> z(n);
  int l = 0, r = 1;
  for (int i = 1; i < n; ++i) {
    z[i] = max(0, min(r - i, z[i - l]));
    while (i + z[i] < n && (s[i + z[i]] == s[z[i]])) ++z[i];
    if (r < i + z[i]) {
      l = i;
      r = i + z[i];
    }
  }
  return z;
}
pair<int, int> euc(const int& m, const int& n) {
  if (n == 0) return pair<int, int>((m >= 0) ? 1 : -1, 0);
  int t = m / n;
  pair<int, int> ans1 = euc(n, m - t * n);
  return pair<int, int>(ans1.second, ans1.first - ans1.second * t);
}
int prod(const int& a, const int& b, const int& M) {
  return ((long long)(a)) * b % M;
}
int sum(const int& a, const int& b, const int& M) {
  int c = a + b;
  return c >= M ? c - M : c;
}
int raz(const int& a, const int& b, const int& M) {
  int c = a - b;
  return c < 0 ? c + M : c;
}
const int LITTLE_BORDER = 400;
struct factorizator {
  int N;
  vector<int> pr;
  vector<int> md;
  vector<int> pw;
  vector<int> without_md;
  factorizator(const int& n) : N(n), md(n), pw(n), without_md(n, 1) {
    for (int i = 2; i < N; ++i) md[i] = i;
    for (int i = 2; i < N; ++i) {
      if (md[i] == i) pr.push_back(i);
      bool worth = true;
      for (int j = 0; worth && (j < ((int)(pr.size()))) && (pr[j] <= md[i]);
           ++j) {
        long long temp = ((long long)(pr[j])) * i;
        if (temp < N)
          md[((int)temp)] = pr[j];
        else
          worth = false;
      }
    }
    for (int i = 2; i < N; ++i) {
      int t = md[i], s = i / t;
      if (md[s] == t) {
        pw[i] = 1 + pw[s];
        without_md[i] = without_md[s];
      } else {
        pw[i] = 1;
        without_md[i] = s;
      }
    }
  }
  void known_factorization(int n, vector<pair<unsigned long long, int>>& v) {
    while (n > 1) {
      v.emplace_back(md[n], pw[n]);
      n = without_md[n];
    }
  }
  vector<pair<unsigned long long, int>> pollardRho(const unsigned long long& n);
  vector<pair<unsigned long long, int>> factor(unsigned long long n);
  template <typename T>
  vector<pair<T, int>> merge(const vector<pair<T, int>>& p1,
                             const vector<pair<T, int>>& p2) {
    vector<pair<T, int>> ans;
    int m = ((int)(p1.size()));
    int n = ((int)(p2.size()));
    int i = 0;
    int j = 0;
    while ((i < m) || (j < n)) {
      if (i < m) {
        if (j < n) {
          if (p1[i].first < p2[j].first)
            ans.push_back(p1[i++]);
          else if (p1[i].first > p2[j].first)
            ans.push_back(p2[j++]);
          else {
            ans.emplace_back(p1[i].first, p1[i].second + p2[j].second);
            ++i;
            ++j;
          }
        } else
          ans.push_back(p1[i++]);
      } else
        ans.push_back(p2[j++]);
    }
    return ans;
  }
};
factorizator fac(LITTLE_BORDER);
unsigned long long experimental_prodll(const unsigned long long& a,
                                       const unsigned long long& b,
                                       const unsigned long long& M,
                                       const int& helper) {
  if (M == 0) return 0;
  unsigned long long q = ((long double)a) * b / M;
  unsigned long long a1 = a * b - q * M;
  unsigned long long a2 = a1;
  int rm = M % helper;
  int ra = a % helper;
  int rb = b % helper;
  int rq = q % helper;
  int r1 = raz(prod(ra, rb, helper), prod(rq, rm, helper), helper);
  int r2 = r1;
  while (true) {
    if (a1 % helper == r1 && a1 < M) return a1;
    a1 -= M;
    r1 = raz(r1, rm, helper);
    a2 += M;
    r2 = sum(r2, rm, helper);
    if (a2 % helper == r2 && a2 < M) return a2;
  }
}
inline int find_helper(const unsigned long long& M) {
  int i = 30;
  while (M % fac.pr[i] == 0) ++i;
  return fac.pr[i];
}
unsigned long long prodll(const unsigned long long& a,
                          const unsigned long long& b,
                          const unsigned long long& M) {
  if (M == 0) return 0;
  unsigned long long helper = find_helper(M);
  return experimental_prodll(a, b, M, helper);
}
unsigned long long sumll(const unsigned long long& a,
                         const unsigned long long& b,
                         const unsigned long long& M) {
  if (a < M - b)
    return a + b;
  else
    return a + b - M;
}
unsigned long long razll(const unsigned long long& a,
                         const unsigned long long& b,
                         const unsigned long long& M) {
  if (a >= b)
    return a - b;
  else
    return a + M - b;
}
template <typename T>
T diff(const T& a, const T& b) {
  return (a > b) ? (a - b) : (b - a);
}
template <typename A>
bool angdis_cmp(const pair<A, A>& a, const pair<A, A>& b) {
  A p;
  if (p = a % b)
    return (p > 0);
  else
    return sq_len(a) < sq_len(b);
}
template <typename T>
int find_min_idx(const vector<T>& v) {
  int ans = 0, n = ((int)(v.size()));
  for (int i = 1; i < n; ++i)
    if (v[i] < v[ans]) ans = i;
  return ans;
}
template <typename T>
vector<int> convex_hull(vector<pair<T, T>>& a) {
  int n = ((int)(a.size()));
  if (n) {
    int mt = find_min_idx(a);
    pair<T, T> d = a[mt];
    for (int i = 0; i < n; ++i) a[i] -= d;
    vector<int> idx(n);
    for (int i = 0; i < n; ++i) idx[i] = i;
    sort(idx.begin(), idx.end(),
         [&](const int& l, const int& r) { return angdis_cmp(a[l], a[r]); });
    vector<int> h(1, idx.front());
    for (auto it = idx.begin() + 1; it != idx.end(); ++it) {
      auto temp = a.begin() + *it;
      if (((int)(h.size())) >= 2) {
        if (are_parallel(a[h.back()], *temp)) h.pop_back();
        while ((((int)(h.size())) >= 3) &&
               npocs(a[h[((int)(h.size())) - 2]], a[h.back()], *temp))
          h.pop_back();
      }
      h.push_back(*it);
    }
    for (int i = 0; i < n; ++i) a[i] += d;
    return h;
  } else
    return vector<int>();
}
pair<int, int> cool_gcd(const int& a, const int& b) {
  if (b) {
    int c = a / b;
    pair<int, int> ans1 = cool_gcd(b, a - b * c);
    return pair<long long, long long>(ans1.second,
                                      ans1.first - ans1.second * c);
  } else
    return pair<int, int>(1, 0);
}
pair<long long, long long> cool_gcdll(const long long& a, const long long& b) {
  if (b) {
    long long c = a / b;
    pair<long long, long long> ans1 = cool_gcdll(b, a - b * c);
    return pair<long long, long long>(ans1.second,
                                      ans1.first - ans1.second * c);
  } else
    return pair<long long, long long>(1ll, 0ll);
}
template <typename T>
T gcd(T a, T b) {
  while (b) {
    a %= b;
    swap(a, b);
  }
  return a;
}
long long pr_p(const long long& a, const long long& b, const long long& p) {
  if (b < 1000000) return (a * b) % p;
  long long temp = pr_p(a, b >> 1ll, p);
  temp = sumll(temp, temp, p);
  if (b & 1ll)
    return sumll(temp, a, p);
  else
    return temp;
}
long long po_p(const long long& a, const long long& b, const long long& p) {
  if (b < 2) {
    if (b == 0)
      return 1;
    else
      return a;
  }
  long long temp = po_p(a, b >> 1ll, p);
  temp = pr_p(temp, temp, p);
  if (b & 1ll)
    return pr_p(temp, a, p);
  else
    return temp;
}
int pow_mod(const int& a, const int& b, const int& p) {
  if (b < 2) {
    if (b == 0)
      return 1;
    else
      return a;
  }
  int temp = pow_mod(a, b >> 1, p);
  temp = prod(temp, temp, p);
  if (b & 1)
    return prod(temp, a, p);
  else
    return temp;
}
long long pow_modll(const long long& a, const long long& b,
                    const long long& p) {
  if (b < 2) {
    if (b == 0)
      return 1;
    else
      return a;
  }
  long long temp = pow_modll(a, b >> 1ll, p);
  temp = prodll(temp, temp, p);
  if (b & 1ll)
    return prodll(temp, a, p);
  else
    return temp;
}
int inverse(int a, int n) {
  int c = cool_gcd(a, n).first;
  if (c < 0) c += n;
  return c;
}
long long inversell(long long a, long long n) {
  long long c = cool_gcdll(a, n).first;
  if (c < 0) c += n;
  return c;
}
template <typename A>
pair<int, int> equal_elements(const vector<A>& u, const vector<A>& v) {
  pair<int, int> ans(INT_MAX, INT_MAX);
  int m = ((int)(u.size())), n = ((int)(v.size()));
  vector<int> id_u(m);
  for (int i = 1; i < m; ++i) id_u[i] = i;
  vector<int> id_v(n);
  for (int i = 1; i < n; ++i) id_v[i] = i;
  sort(id_u.begin(), id_u.end(),
       [&](const int& x, const int& y) { return u[x] < u[y]; });
  sort(id_v.begin(), id_v.end(),
       [&](const int& x, const int& y) { return v[x] < v[y]; });
  int i = 0;
  int j = 0;
  while ((i < m) && (j < n)) {
    if (u[id_u[i]] < v[id_v[j]])
      ++i;
    else if (v[id_v[j]] < u[id_u[i]])
      ++j;
    else {
      ans = min(ans, pair<int, int>(id_v[j], id_u[i]));
      ++j;
    }
  }
  if (ans.first == INT_MAX)
    return pair<int, int>(-1, -1);
  else
    return pair<int, int>(ans.second, ans.first);
}
long long discr_log(long long a, long long b, long long n) {
  int k = ((int)(sqrt((long double)n)));
  long long a1 = inversell(a, n);
  int l = k + 20;
  long long a2 = po_p(a1, k, n);
  vector<long long> seq1(k), seq2(l);
  seq1.front() = 1;
  for (int i = 1; i < k; ++i) seq1[i] = pr_p(seq1[i - 1], a, n);
  seq2.front() = b;
  for (int i = 1; i < l; ++i) seq2[i] = pr_p(seq2[i - 1], a2, n);
  pair<long long, long long> e = equal_elements(seq1, seq2);
  if (e.first == -1)
    return -1;
  else
    return e.first + e.second * k;
}
long long common_discr_log(long long a, long long b, long long n) {
  const int C = 70;
  a %= n;
  b %= n;
  if (gcd(n, a) != 1) {
    for (int i = 0; i < C; ++i) {
      if (po_p(a, i, n) == b) {
        return i;
      }
    }
    long long multp = po_p(a, C, n);
    long long g = gcd(multp, n);
    if (b % g) {
      return -1;
    } else {
      b /= g;
      n /= g;
      multp /= g;
      long long pop_back = inversell(multp, n);
      b = pr_p(b, pop_back, n);
      long long ans = discr_log(a, b, n);
      if (ans == -1)
        return -1;
      else
        return ans + C;
    }
  } else
    return discr_log(a, b, n);
}
const unsigned long long minimum_composites[] = {1ull,
                                                 2046ull,
                                                 1373652ull,
                                                 25326000ull,
                                                 3215031750ull,
                                                 2152302898746ull,
                                                 3474749660382ull,
                                                 341550071728320ull,
                                                 341550071728320ull,
                                                 3825123056546413050ull,
                                                 3825123056546413050ull,
                                                 18446744073709551615ull};
const int little_primes[] = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37};
bool miller_rabin_check(const unsigned long long& n,
                        const unsigned long long& prime,
                        const unsigned long long& odd,
                        const unsigned long long& pot,
                        const unsigned long long& nmo) {
  long long x = pow_modll(prime, odd, n);
  if ((x == nmo) || (x == 1)) return true;
  for (int i = 1; i < pot; ++i) {
    x = prodll(x, x, n);
    if (x == nmo) return true;
    if (x == 1) return false;
  }
  return false;
}
bool is_prime(const unsigned long long& n) {
  if (n <= 2) return (n == 2);
  unsigned long long odd, pot, nmo;
  nmo = n - 1;
  odd = nmo;
  pot = 0;
  while ((odd & 1ll) == 0) {
    odd >>= 1ll;
    ++pot;
  }
  int i = 0;
  while (n > minimum_composites[i]) {
    if (miller_rabin_check(n, little_primes[i], odd, pot, nmo) == false)
      return false;
    ++i;
  }
  return true;
}
vector<pair<unsigned long long, int>> factorizator::pollardRho(
    const unsigned long long& n) {
  if (is_prime(n))
    return vector<pair<unsigned long long, int>>(
        1, pair<unsigned long long, int>(n, 1));
  int step = 1;
  while (true) {
    step <<= 1;
    unsigned long long a = rng();
    unsigned long long b = a;
    unsigned long long d = 1;
    for (int j = 0; j < step; ++j) {
      a = sumll(1, prodll(a, a, n), n);
      b = sumll(1, prodll(b, b, n), n);
      b = sumll(1, prodll(b, b, n), n);
      d = gcd(diff(a, b), n);
      if (d > 1) {
        break;
      }
    }
    if (1 < d && d < n) return merge(pollardRho(d), pollardRho(n / d));
  }
  return vector<pair<unsigned long long, int>>();
}
vector<pair<unsigned long long, int>> factorizator::factor(
    unsigned long long n) {
  vector<pair<unsigned long long, int>> ans;
  for (int i = 0; n >= N && i < ((int)(pr.size())) && pr[i] < LITTLE_BORDER;
       ++i)
    while (n % pr[i] == 0) {
      if (((int)(ans.size())) && ans.back().first == pr[i])
        ++ans.back().second;
      else
        ans.emplace_back(pr[i], 1);
      n /= pr[i];
    }
  if (n < N) {
    known_factorization(n, ans);
    return ans;
  } else
    return merge(ans, pollardRho(n));
}
template <typename T>
T phi(const vector<pair<T, int>>& v) {
  T ans = 1;
  for (int i = 0; i < ((int)(v.size())); ++i) {
    ans *= v[i].first - 1;
    for (int j = 1; j < v[i].second; ++j) ans *= v[i].first;
  }
  return ans;
}
unsigned long long phi(const unsigned long long& n) {
  return phi(fac.factor(n));
}
bool check_primitive_root(const int& ph, const vector<int>& to_check,
                          const int& r, const int& n) {
  for (int i = 0; i < ((int)(to_check.size())); ++i)
    if (pow_mod(r, to_check[i], n) == 1) return false;
  return (pow_mod(r, ph, n) == 1);
}
int primitive_root(const int& n) {
  if (n < 3) return n - 1;
  int p = phi(n);
  vector<pair<unsigned long long, int>> f = fac.factor(p);
  vector<int> to_check(((int)(f.size())));
  for (int i = 0; i < ((int)(f.size())); ++i) to_check[i] = p / f[i].first;
  for (int i = 2; i < n; ++i)
    if (check_primitive_root(p, to_check, i, n)) return i;
  return -1;
}
int unite_mod(const int& a, const int& p, const int& b, const int& q) {
  pair<int, int> c = cool_gcd(p, q);
  int pr = p * q;
  int ans = ((a * c.second * q + b * c.first * p) % pr + pr) % pr;
  return ans;
}
long long unite_modll(const long long& a, const long long& p,
                      const long long& b, const long long& q) {
  pair<long long, long long> c = cool_gcdll(p, q);
  long long pr = p * q;
  long long ans = ((a * c.second * q + b * c.first * p) % pr + pr) % pr;
  return ans;
}
pair<int, int> power_v(int n, const int& p) {
  int ans = 0;
  while (n % p == 0) {
    n /= p;
    ++ans;
  }
  return pair<int, int>(ans, n);
}
int square_root_prime_modulo(int c, int n, const int& pr, const int& k) {
  c %= n;
  if (c) {
    pair<int, int> kek = power_v(c, pr);
    int l = kek.first;
    if (l & 1) return -1;
    if (l > 0) {
      int pwl = 1;
      for (int i = 0; i < l; ++i) pwl *= pr;
      n /= pwl;
      c /= pwl;
      int ans1 = square_root_prime_modulo(c, n, pr, k - l);
      if (ans1 == -1) return -1;
      for (int i = 0; i < (l >> 1); ++i) ans1 *= pr;
      return ans1;
    } else {
      int primitive;
      if (n & 1)
        primitive = primitive_root(n);
      else
        primitive = 5;
      int u = ((int)discr_log(primitive, c, n));
      if (u == -1) return -1;
      if (u & 1) return -1;
      return pow_mod(primitive, u >> 1, n);
    }
  } else
    return 0;
}
int square_root_modulo(const int& c, const int& n) {
  vector<pair<unsigned long long, int>> f = fac.factor(n);
  int a = 0, p = 1;
  for (int i = 0; i < ((int)(f.size())); ++i) {
    int q = 1;
    for (int j = 0; j < f[i].second; ++j) q *= f[i].first;
    int b = square_root_prime_modulo(c, q, f[i].first, f[i].second);
    if (b == -1) return -1;
    a = unite_mod(a, p, b, q);
    p *= q;
  }
  return a;
}
namespace BGF {
inline int cntd(unsigned x) {
  int ans = 0;
  while (x) {
    if (x & 1) ++ans;
    x >>= 1;
  }
  return ans;
}
inline long long cntdll(unsigned long long x) {
  long long ans = 0;
  while (x) {
    if (x & 1) ++ans;
    x >>= 1;
  }
  return ans;
}
inline int clzd(unsigned x) {
  int ans = 32;
  while (x) {
    --ans;
    x >>= 1;
  }
  return ans;
}
inline long long clzdll(unsigned long long x) {
  long long ans = 64;
  while (x) {
    --ans;
    x >>= 1;
  }
  return ans;
}
inline int ctzd(unsigned x) {
  if (x == 0) return 32;
  int ans = 0;
  while ((x & 1) == 0) {
    ++ans;
    x >>= 1;
  }
  return ans;
}
inline long long ctzdll(unsigned long long x) {
  if (x == 0) return 64;
  long long ans = 0;
  while ((x & 1) == 0) {
    ++ans;
    x >>= 1;
  }
  return ans;
}
}  // namespace BGF
namespace BGF {
const unsigned BHC = 65535u;
const unsigned long long BHCL = 65535ull;
const unsigned BHN = 65536u;
const unsigned BHL = 16u;
const unsigned LH = 4294901760u;
const unsigned RH = 65535u;
const unsigned long long BHLL1 = 16ull;
const unsigned long long BHLL2 = 32ull;
const unsigned long long BHLL3 = 48ull;
const unsigned long long LHL = 18446744069414584320ull;
const unsigned long long RHL = 4294967295ull;
int CNT[BHN];
int CLZ[BHN];
int CTZ[BHN];
inline void initialize() {
  CNT[0] = 0;
  CLZ[0] = BHL + BHL;
  CTZ[0] = BHL;
  for (unsigned i = 1; i < BHN; ++i) {
    CNT[i] = cntd(i);
    CLZ[i] = clzd(i);
    CTZ[i] = ctzd(i);
  }
}
inline int cntc(const unsigned& x) {
  return CNT[x & BHC] + CNT[(x >> BHL) & BHC];
}
inline long long cntcll(const unsigned long long& x) {
  return CNT[x & BHCL] + CNT[(x >> BHLL1) & BHCL] + CNT[(x >> BHLL2) & BHCL] +
         CNT[(x >> BHLL3) & BHCL];
}
inline int clzc(const unsigned& x) {
  if (x & LH)
    return CLZ[x >> BHL] - BHL;
  else
    return CLZ[x];
}
inline long long clzcll(const unsigned long long& x) {
  if (x & LHL)
    return clzc(x >> BHLL2);
  else
    return clzc(x) + BHLL2;
}
inline int ctzc(const unsigned& x) {
  if (x & RH)
    return CTZ[x & RH];
  else
    return CTZ[x >> BHL] + BHL;
}
inline long long ctzcll(const unsigned long long& x) {
  if (x & RHL)
    return ctzc(x);
  else
    return ctzc(x >> BHLL2) + BHLL2;
}
}  // namespace BGF
const string INPUT_FILE = "input.txt";
const string OUTPUT_FILE = "output.txt";
void relax(long long& a, const long long& b) { a = max(a, b); }
int another_main() {
  int n, k;
  cin >> n >> k;
  vector<int> a(k);
  cin >> a;
  --a;
  int N = 1 << n;
  vector<bool> v(N);
  for (int i : a) v[i] = true;
  N /= 2;
  vector<vector<long long>> dp(N, vector<long long>(4, -1e9));
  for (int i = 0; i < N; ++i) {
    int x = v[2 * i];
    int y = v[2 * i + 1];
    int dob = (x || y);
    dp[i][2 * x + y] = dob;
    dp[i][2 * y + x] = dob;
  }
  while (N > 1) {
    N /= 2;
    vector<vector<long long>> dq(N, vector<long long>(4, -1e9));
    for (int i = 0; i < N; ++i)
      for (int j = 0; j < 4; ++j)
        for (int l = 0; l < 4; ++l) {
          int base11 = (j & l & 1);
          int base1 = (((j & 1) | (l & 1)) != 0);
          int base2 = (((j & 2) | (l & 2)) != 0);
          long long base3 = dp[2 * i][j] + dp[2 * i + 1][l];
          long long base = base1 + base2 + base3;
          int p = -3;
          if (base2 || base11)
            p = 2 | base1;
          else if (base2 == 0 && base1 == 0)
            p = 0;
          if (p != -3) {
            relax(dq[i][p], base + (p != 0));
            if (dq[i][p] == 5) {
              i = i;
            }
          } else {
            relax(dq[i][1], base);
            relax(dq[i][2], base + 1);
          }
        }
    dp = dq;
  }
  long long base0 = dp[0][0];
  long long base1 = dp[0][1] + 1;
  long long base2 = dp[0][2] + 1;
  long long base3 = dp[0][3] + 1;
  cout << max(max(base0, base1), max(base2, base3)) << '\n';
  return 0;
}
int main() {
  ios_base::sync_with_stdio(0);
  cin.tie(0);
  cout.tie(0);
  cerr.tie(0);
  BGF::initialize();
  int AMR = another_main();
  return AMR;
}

```