#include <bits/stdc++.h>
using namespace std;
const int N = 300000 + 1;
const int inf = (int)1e9 + 1;
const long long big = (long long)1e18 + 1;
const int P = 239;
const int MOD = (int)1e9 + 9;
const int MOD1 = (int)1e9 + 9;
const double eps = 1e-9;
const double pi = atan2(0, -1);
const int ABC = 26;
const int B = 555;
long long fib[400000];
void init() {
  fib[1] = 1;
  fib[2] = 1;
  for (int i = 2; i < 400000; i++) {
    fib[i] = fib[i - 1] + fib[i - 2];
    if (fib[i] >= MOD) fib[i] -= MOD;
  }
}
int n;
long long a[N], pref[N], add[N];
vector<pair<int, int> > temp;
void update() {
  fill(add, add + n, 0);
  for (auto it : temp) {
    int l = it.first, r = it.second;
    add[l]++;
    if (add[l] >= MOD) add[l] -= MOD;
    if (r + 1 < n) {
      add[r + 1] -= fib[(r - l + 1) + 1];
      if (add[r + 1] < 0) add[r + 1] += MOD;
    }
    if (r + 2 < n) {
      add[r + 2] -= fib[(r - l + 1)];
      if (add[r + 2] < 0) add[r + 2] += MOD;
    }
  }
  temp.clear();
  for (int i = 1; i < n; i++) {
    add[i] += add[i - 1];
    if (add[i] >= MOD) add[i] -= MOD;
    if (i > 1) {
      add[i] += add[i - 2];
      if (add[i] >= MOD) add[i] -= MOD;
    }
  }
  for (int i = 0; i < n; i++) {
    a[i] += add[i];
    if (a[i] >= MOD) a[i] -= MOD;
  }
  pref[0] = 0;
  for (int i = 0; i < n; i++) {
    pref[i + 1] = pref[i] + a[i];
    if (pref[i + 1] >= MOD) pref[i + 1] -= MOD;
  }
}
long long fib_sum(int x) {
  long long res = fib[x + 2] - 1;
  if (res < 0) res += MOD;
  return res;
}
int main() {
  ios_base::sync_with_stdio(0);
  cin.tie(0);
  cout.precision(20);
  init();
  int m;
  cin >> n >> m;
  for (int i = 0; i < n; i++) cin >> a[i];
  update();
  for (int i = 0; i < m; i++) {
    int t, l, r;
    cin >> t >> l >> r;
    l--;
    r--;
    if (t == 1) {
      temp.push_back({l, r});
      if ((int)temp.size() > B) update();
    } else {
      long long ans = pref[r + 1] - pref[l];
      if (ans < 0) ans += MOD;
      for (auto it : temp)
        if (max(l, it.first) <= min(r, it.second)) {
          ans += fib_sum(min(r, it.second) - it.first + 1);
          if (ans >= MOD) ans -= MOD;
          ans -= fib_sum(max(l, it.first) - it.first);
          if (ans < 0) ans += MOD;
        }
      cout << ans << "\n";
    }
  }
  return 0;
}

### Prompt

Generate a CPP solution to the following problem:

In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation 

F1 = 1; F2 = 1; Fn = Fn - 1 + Fn - 2 (n > 2).

DZY loves Fibonacci numbers very much. Today DZY gives you an array consisting of n integers: a1, a2, ..., an. Moreover, there are m queries, each query has one of the two types:

  1. Format of the query "1 l r". In reply to the query, you need to add Fi - l + 1 to each element ai, where l ≤ i ≤ r. 
  2. Format of the query "2 l r". In reply to the query you should output the value of <image> modulo 1000000009 (109 + 9). 



Help DZY reply to all the queries.

Input

The first line of the input contains two integers n and m (1 ≤ n, m ≤ 300000). The second line contains n integers a1, a2, ..., an (1 ≤ ai ≤ 109) — initial array a.

Then, m lines follow. A single line describes a single query in the format given in the statement. It is guaranteed that for each query inequality 1 ≤ l ≤ r ≤ n holds.

Output

For each query of the second type, print the value of the sum on a single line.

Examples

Input

4 4
1 2 3 4
1 1 4
2 1 4
1 2 4
2 1 3


Output

17
12

Note

After the first query, a = [2, 3, 5, 7].

For the second query, sum = 2 + 3 + 5 + 7 = 17.

After the third query, a = [2, 4, 6, 9].

For the fourth query, sum = 2 + 4 + 6 = 12.

### Response

```cpp
#include <bits/stdc++.h>
using namespace std;
const int N = 300000 + 1;
const int inf = (int)1e9 + 1;
const long long big = (long long)1e18 + 1;
const int P = 239;
const int MOD = (int)1e9 + 9;
const int MOD1 = (int)1e9 + 9;
const double eps = 1e-9;
const double pi = atan2(0, -1);
const int ABC = 26;
const int B = 555;
long long fib[400000];
void init() {
  fib[1] = 1;
  fib[2] = 1;
  for (int i = 2; i < 400000; i++) {
    fib[i] = fib[i - 1] + fib[i - 2];
    if (fib[i] >= MOD) fib[i] -= MOD;
  }
}
int n;
long long a[N], pref[N], add[N];
vector<pair<int, int> > temp;
void update() {
  fill(add, add + n, 0);
  for (auto it : temp) {
    int l = it.first, r = it.second;
    add[l]++;
    if (add[l] >= MOD) add[l] -= MOD;
    if (r + 1 < n) {
      add[r + 1] -= fib[(r - l + 1) + 1];
      if (add[r + 1] < 0) add[r + 1] += MOD;
    }
    if (r + 2 < n) {
      add[r + 2] -= fib[(r - l + 1)];
      if (add[r + 2] < 0) add[r + 2] += MOD;
    }
  }
  temp.clear();
  for (int i = 1; i < n; i++) {
    add[i] += add[i - 1];
    if (add[i] >= MOD) add[i] -= MOD;
    if (i > 1) {
      add[i] += add[i - 2];
      if (add[i] >= MOD) add[i] -= MOD;
    }
  }
  for (int i = 0; i < n; i++) {
    a[i] += add[i];
    if (a[i] >= MOD) a[i] -= MOD;
  }
  pref[0] = 0;
  for (int i = 0; i < n; i++) {
    pref[i + 1] = pref[i] + a[i];
    if (pref[i + 1] >= MOD) pref[i + 1] -= MOD;
  }
}
long long fib_sum(int x) {
  long long res = fib[x + 2] - 1;
  if (res < 0) res += MOD;
  return res;
}
int main() {
  ios_base::sync_with_stdio(0);
  cin.tie(0);
  cout.precision(20);
  init();
  int m;
  cin >> n >> m;
  for (int i = 0; i < n; i++) cin >> a[i];
  update();
  for (int i = 0; i < m; i++) {
    int t, l, r;
    cin >> t >> l >> r;
    l--;
    r--;
    if (t == 1) {
      temp.push_back({l, r});
      if ((int)temp.size() > B) update();
    } else {
      long long ans = pref[r + 1] - pref[l];
      if (ans < 0) ans += MOD;
      for (auto it : temp)
        if (max(l, it.first) <= min(r, it.second)) {
          ans += fib_sum(min(r, it.second) - it.first + 1);
          if (ans >= MOD) ans -= MOD;
          ans -= fib_sum(max(l, it.first) - it.first);
          if (ans < 0) ans += MOD;
        }
      cout << ans << "\n";
    }
  }
  return 0;
}

```