#include <bits/stdc++.h>
using namespace std;
inline int readint() {
  int x = 0;
  bool f = 0;
  char c = getchar();
  while (!isdigit(c) && c != '-') c = getchar();
  if (c == '-') {
    f = 1;
    c = getchar();
  }
  while (isdigit(c)) {
    x = x * 10 + c - '0';
    c = getchar();
  }
  return f ? -x : x;
}
const int maxn = 1e5 + 5;
int n, m = 0, a[maxn];
const long long mod = 998244353, g = 3, invg = 332748118, inv2 = 499122177;
long long ksm(long long a, long long b) {
  long long ans = 1;
  while (b) {
    if (b % 2 == 1) ans = ans * a % mod;
    a = a * a % mod;
    b /= 2;
  }
  return ans;
}
int rev[maxn * 2];
long long w[maxn * 2], w2[maxn * 2], invn;
void init(int n) {
  for (int i = 1; i < n; i++)
    rev[i] = rev[i >> 1] >> 1 | ((i & 1) ? n >> 1 : 0);
  for (int i = 1; i < n; i *= 2) {
    w[i] = 1;
    long long wn = ksm(g, (mod - 1) / (i * 2));
    for (int j = 1; j < i; j++) w[i + j] = w[i + j - 1] * wn % mod;
  }
  for (int i = 1; i < n; i *= 2) {
    w2[i] = 1;
    long long wn = ksm(invg, (mod - 1) / (i * 2));
    for (int j = 1; j < i; j++) w2[i + j] = w2[i + j - 1] * wn % mod;
  }
  invn = ksm(n, mod - 2);
}
vector<long long> FFT(vector<long long> F, bool flag) {
  int n = F.size();
  for (int i = 0; i < n; i++)
    if (i < rev[i]) swap(F[i], F[rev[i]]);
  for (int i = 1; i < n; i *= 2)
    for (int j = 0; j < n; j += i * 2)
      for (int k = j; k < j + i; k++) {
        long long t = F[k + i] * (flag ? w[i + k - j] : w2[i + k - j]) % mod;
        F[k + i] = (F[k] - t + mod) % mod;
        F[k] = (F[k] + t) % mod;
      }
  if (!flag)
    for (int i = 0; i < n; i++) F[i] = F[i] * invn % mod;
  return F;
}
vector<vector<vector<long long>>> solve(int l, int r) {
  if (l == r) {
    if (a[r] == 1)
      return {{{0, 0}, {0, 0}}, {{0, 0}, {0, 1}}};
    else
      return {{{0, 2}, {0, 0}}, {{0, 0}, {0, 0}}};
  }
  int mid = l + (r - l) / 2;
  vector<vector<vector<long long>>> F = solve(l, mid), G = solve(mid + 1, r),
                                    H = {{{}, {}}, {{}, {}}}, A = F, B = G;
  for (int i = 0; i < 2; i++) {
    A[0][i][1] = (A[0][i][1] + A[1][i][1]) % mod;
    A[1][i][1] = 0;
    B[i][0][1] = (B[i][0][1] + B[i][1][1]) % mod;
    B[i][1][1] = 0;
  }
  int len = 1;
  while (len < r - l + 2) len *= 2;
  init(len);
  for (int i = 0; i < 2; i++)
    for (int j = 0; j < 2; j++) {
      F[i][j].resize(len);
      G[i][j].resize(len);
      F[i][j] = FFT(F[i][j], 1);
      G[i][j] = FFT(G[i][j], 1);
      if (r - l <= 2) {
        A[i][j].resize(len);
        B[i][j].resize(len);
        A[i][j] = FFT(A[i][j], 1);
        B[i][j] = FFT(B[i][j], 1);
      }
    }
  for (int i = 0; i < 2; i++)
    for (int j = 0; j < 2; j++) {
      H[i][j].resize(r - l + 2);
      for (int x = 0; x < 2; x++)
        for (int y = 0; y < 2; y++) {
          vector<long long> C(len);
          for (int k = 0; k < len; k++) C[k] = F[i][x][k] * G[y][j][k] % mod;
          C = FFT(C, 0);
          for (int k = 1; k <= r - l + 1; k++)
            H[i][j][k] = (H[i][j][k] + C[k]) % mod;
          if (r - l <= 2) {
            for (int k = 0; k < len; k++) C[k] = A[i][x][k] * B[y][j][k] % mod;
            C = FFT(C, 0);
          }
          for (int k = 1; k <= r - l; k++) {
            long long res = C[k + 1];
            if (x && y) res = res * 2 % mod;
            if (!x && !y) res = res * inv2 % mod;
            H[i][j][k] = (H[i][j][k] + res) % mod;
          }
        }
    }
  return H;
}
long long fac[maxn];
int main() {
  n = readint();
  while (n) {
    a[++m] = readint();
    if (n < a[m]) {
      printf("0\n");
      return 0;
    }
    for (int i = 1; i < a[m]; i++)
      if (readint() != a[m]) {
        printf("0\n");
        return 0;
      }
    n -= a[m];
  }
  fac[0] = 1;
  for (int i = 1; i <= m; i++) fac[i] = fac[i - 1] * i % mod;
  vector<vector<vector<long long>>> F = solve(1, m);
  long long ans = 0;
  for (int i = 1; i <= m; i++) {
    long long res = 0;
    for (int j = 0; j < 2; j++)
      for (int k = 0; k < 2; k++) res = (res + F[j][k][i]) % mod;
    ans = (ans + res * fac[i] % mod * ((m - i) % 2 == 0 ? 1 : mod - 1) % mod) %
          mod;
  }
  printf("%lld\n", ans);
  return 0;
}

### Prompt

Please formulate a Cpp solution to the following problem:

For a permutation p of numbers 1 through n, we define a stair array a as follows: a_i is length of the longest segment of permutation which contains position i and is made of consecutive values in sorted order: [x, x+1, …, y-1, y] or [y, y-1, …, x+1, x] for some x ≤ y. For example, for permutation p = [4, 1, 2, 3, 7, 6, 5] we have a = [1, 3, 3, 3, 3, 3, 3]. 

You are given the stair array a. Your task is to calculate the number of permutations which have stair array equal to a. Since the number can be big, compute it modulo 998 244 353. Note that this number can be equal to zero.

Input

The first line of input contains integer n (1 ≤ n ≤ 10^5) — the length of a stair array a.

The second line of input contains n integers a_1, a_2, …, a_n (1 ≤ a_i ≤ n).

Output

Print the number of permutations which have stair array equal to a. Since the number can be big, compute it modulo 998 244 353.

Examples

Input


6
3 3 3 1 1 1


Output


6


Input


7
4 4 4 4 3 3 3


Output


6


Input


1
1


Output


1


Input


8
2 2 2 2 2 2 1 1


Output


370


Input


4
3 2 3 1


Output


0

### Response

```cpp
#include <bits/stdc++.h>
using namespace std;
inline int readint() {
  int x = 0;
  bool f = 0;
  char c = getchar();
  while (!isdigit(c) && c != '-') c = getchar();
  if (c == '-') {
    f = 1;
    c = getchar();
  }
  while (isdigit(c)) {
    x = x * 10 + c - '0';
    c = getchar();
  }
  return f ? -x : x;
}
const int maxn = 1e5 + 5;
int n, m = 0, a[maxn];
const long long mod = 998244353, g = 3, invg = 332748118, inv2 = 499122177;
long long ksm(long long a, long long b) {
  long long ans = 1;
  while (b) {
    if (b % 2 == 1) ans = ans * a % mod;
    a = a * a % mod;
    b /= 2;
  }
  return ans;
}
int rev[maxn * 2];
long long w[maxn * 2], w2[maxn * 2], invn;
void init(int n) {
  for (int i = 1; i < n; i++)
    rev[i] = rev[i >> 1] >> 1 | ((i & 1) ? n >> 1 : 0);
  for (int i = 1; i < n; i *= 2) {
    w[i] = 1;
    long long wn = ksm(g, (mod - 1) / (i * 2));
    for (int j = 1; j < i; j++) w[i + j] = w[i + j - 1] * wn % mod;
  }
  for (int i = 1; i < n; i *= 2) {
    w2[i] = 1;
    long long wn = ksm(invg, (mod - 1) / (i * 2));
    for (int j = 1; j < i; j++) w2[i + j] = w2[i + j - 1] * wn % mod;
  }
  invn = ksm(n, mod - 2);
}
vector<long long> FFT(vector<long long> F, bool flag) {
  int n = F.size();
  for (int i = 0; i < n; i++)
    if (i < rev[i]) swap(F[i], F[rev[i]]);
  for (int i = 1; i < n; i *= 2)
    for (int j = 0; j < n; j += i * 2)
      for (int k = j; k < j + i; k++) {
        long long t = F[k + i] * (flag ? w[i + k - j] : w2[i + k - j]) % mod;
        F[k + i] = (F[k] - t + mod) % mod;
        F[k] = (F[k] + t) % mod;
      }
  if (!flag)
    for (int i = 0; i < n; i++) F[i] = F[i] * invn % mod;
  return F;
}
vector<vector<vector<long long>>> solve(int l, int r) {
  if (l == r) {
    if (a[r] == 1)
      return {{{0, 0}, {0, 0}}, {{0, 0}, {0, 1}}};
    else
      return {{{0, 2}, {0, 0}}, {{0, 0}, {0, 0}}};
  }
  int mid = l + (r - l) / 2;
  vector<vector<vector<long long>>> F = solve(l, mid), G = solve(mid + 1, r),
                                    H = {{{}, {}}, {{}, {}}}, A = F, B = G;
  for (int i = 0; i < 2; i++) {
    A[0][i][1] = (A[0][i][1] + A[1][i][1]) % mod;
    A[1][i][1] = 0;
    B[i][0][1] = (B[i][0][1] + B[i][1][1]) % mod;
    B[i][1][1] = 0;
  }
  int len = 1;
  while (len < r - l + 2) len *= 2;
  init(len);
  for (int i = 0; i < 2; i++)
    for (int j = 0; j < 2; j++) {
      F[i][j].resize(len);
      G[i][j].resize(len);
      F[i][j] = FFT(F[i][j], 1);
      G[i][j] = FFT(G[i][j], 1);
      if (r - l <= 2) {
        A[i][j].resize(len);
        B[i][j].resize(len);
        A[i][j] = FFT(A[i][j], 1);
        B[i][j] = FFT(B[i][j], 1);
      }
    }
  for (int i = 0; i < 2; i++)
    for (int j = 0; j < 2; j++) {
      H[i][j].resize(r - l + 2);
      for (int x = 0; x < 2; x++)
        for (int y = 0; y < 2; y++) {
          vector<long long> C(len);
          for (int k = 0; k < len; k++) C[k] = F[i][x][k] * G[y][j][k] % mod;
          C = FFT(C, 0);
          for (int k = 1; k <= r - l + 1; k++)
            H[i][j][k] = (H[i][j][k] + C[k]) % mod;
          if (r - l <= 2) {
            for (int k = 0; k < len; k++) C[k] = A[i][x][k] * B[y][j][k] % mod;
            C = FFT(C, 0);
          }
          for (int k = 1; k <= r - l; k++) {
            long long res = C[k + 1];
            if (x && y) res = res * 2 % mod;
            if (!x && !y) res = res * inv2 % mod;
            H[i][j][k] = (H[i][j][k] + res) % mod;
          }
        }
    }
  return H;
}
long long fac[maxn];
int main() {
  n = readint();
  while (n) {
    a[++m] = readint();
    if (n < a[m]) {
      printf("0\n");
      return 0;
    }
    for (int i = 1; i < a[m]; i++)
      if (readint() != a[m]) {
        printf("0\n");
        return 0;
      }
    n -= a[m];
  }
  fac[0] = 1;
  for (int i = 1; i <= m; i++) fac[i] = fac[i - 1] * i % mod;
  vector<vector<vector<long long>>> F = solve(1, m);
  long long ans = 0;
  for (int i = 1; i <= m; i++) {
    long long res = 0;
    for (int j = 0; j < 2; j++)
      for (int k = 0; k < 2; k++) res = (res + F[j][k][i]) % mod;
    ans = (ans + res * fac[i] % mod * ((m - i) % 2 == 0 ? 1 : mod - 1) % mod) %
          mod;
  }
  printf("%lld\n", ans);
  return 0;
}

```