#include <stdio.h>
#include <string.h>
#include <iostream>
#include <algorithm>
#include <utility>
#include <queue>
using namespace std;
#define rep(i, n) for (int i = 0; i < (int)(n); i++)
#define mp make_pair
#define fst first
#define snd second
typedef long long Int;
#define MOD (1000000009LL)
#define INF (1<<28)
inline void add(Int &a, Int b) { a = (a + b) % MOD; }

#define P (2000001)
Int inv[P], ft[P], ift[P];
const int dx[4] = {1, 0, -1, 0};
const int dy[4] = {0, 1, 0, -1};

int n, m;
pair<int, int> ps[64];
Int fw[64][64];
#define D (128)
#define H (52)
int xof[2][D][D], yof[2][D][D], dt[2][D][D];
Int cnt[2][D][D][2];

Int f(int sx, int sy, int ex, int ey) {
    const int dx = ex-sx, dy = ey-sy;
    return dx >= 0 && dy >= 0 ? ft[dx+dy] * ift[dx] % MOD * ift[dy] % MOD : 0;
}

Int calc(int sx, int sy, int ex, int ey) {
    static Int dp[64][2];
    rep (i, m) dp[i][0] = 0, dp[i][1] = f(sx, sy, ps[i].fst, ps[i].snd);
    Int ans = f(sx, sy, ex, ey);
    rep (i, m) rep (t, 2) {
        const Int w = dp[i][t] * f(ps[i].fst, ps[i].snd, ex, ey);
        if (t) add(ans, MOD-w%MOD); else add(ans, w);
        for (int j = i+1; j < m; j++) add(dp[j][(t+1)%2], dp[i][t]*fw[i][j]);
    }
    return ans;
}

inline bool valid(int w, int L, int x, int y) {
    return 0 <= x && x < L && 0 <= y && y < L && dt[w][x][y] < INF;
}

void build(int w, int L) {
    queue<pair<int, int> > q;
    q.push(mp(0, 0));
    memset(dt[w], -1, sizeof(dt[0]));
    dt[w][0][0] = 0;
    rep (x, L) rep (y, L) {
        if (w == 0) xof[w][x][y] = x, yof[w][x][y] = y;
        else xof[w][x][y] = n-1-x, yof[w][x][y] = n-1-y;
    }
    rep (x, L) rep (y, L) rep (i, m) {
        if (mp(xof[w][x][y], yof[w][x][y]) == ps[i]) dt[w][x][y] = INF;
    }
    while (!q.empty()) {
        const pair<int, int> vv = q.front();
        q.pop();
        const int x = vv.fst, y = vv.snd;
        rep (d, 4) {
            const int nx = x+dx[d], ny = y+dy[d];
            if (valid(w, L, nx, ny) && dt[w][nx][ny] == -1) {
                q.push(mp(nx, ny)), dt[w][nx][ny] = dt[w][x][y] + 1;
            }
        }
    }
    rep (x, L) rep (y, L) if (dt[w][x][y] == -1) dt[w][x][y] = INF;
    vector<pair<int, pair<int, int> > > vs;
    rep (x, L) rep (y, L) if (dt[w][x][y] < INF) {
        vs.push_back(mp(dt[w][x][y], mp(x, y)));
    }
    sort(vs.begin(), vs.end());
    memset(cnt[w], 0, sizeof(cnt[0]));
    cnt[w][0][0][1] = 1;
    rep (i, vs.size()) rep (t, 2) {
        const int x = vs[i].snd.fst, y = vs[i].snd.snd;
        rep (d, 4) {
            const int nx = x+dx[d], ny = y+dy[d];
            if (valid(w, L, nx, ny) && dt[w][nx][ny] == dt[w][x][y]+1) {
                add(cnt[w][nx][ny][d/2], cnt[w][x][y][t]);
            }
        }
    }
}

int main() {
    inv[1] = 1;
    for (int i = 2; i < P; i++) inv[i] = inv[MOD%i] * (MOD - MOD/i) % MOD;
    ft[0] = ift[0] = 1;
    rep (i, P-1) ft[i+1] = (i+1)*ft[i]%MOD, ift[i+1] = inv[i+1]*ift[i]%MOD;
    scanf("%d%d", &n, &m);
    rep (i, m) scanf("%d%d", &ps[i].fst, &ps[i].snd);
    rep (i, m) ps[i].fst--, ps[i].snd--;
    sort(ps, ps+m);
    rep (i, m) rep (j, m) {
        fw[i][j] = f(ps[i].fst, ps[i].snd, ps[j].fst, ps[j].snd);
    }
    if (n < D) {
        build(0, n);
        cout << cnt[0][n-1][n-1][0] << endl;
        return 0;
    }
    build(0, H), build(1, H);
    Int ans = 0;
    int cur = INF;
    vector<pair<int, int> > vs;
    rep (i, H) rep (j, H) if (dt[0][i][j] < INF && cnt[0][i][j][1]) {
        vs.push_back(mp(i, j));
    }
    rep (i, H) rep (j, H) if (dt[1][i][j] < INF && cnt[1][i][j][1]) {
        rep (k, vs.size()) {
            const int x = vs[k].fst, y = vs[k].snd;;
            const int sum = dt[0][x][y] + dt[1][i][j] 
                + xof[1][i][j] - x + yof[1][i][j] - y;
            if (sum > cur) continue;
            const Int w = calc(x, y, xof[1][i][j], yof[1][i][j]);
            if (w == 0) continue;
            if (cur > sum) cur = sum, ans = 0;
            if (cur == sum) add(ans, cnt[0][x][y][1]*w%MOD*cnt[1][i][j][1]);
        }
    }
    cout << ans << endl;
    return 0;
}
### Prompt

Develop a solution in cpp to the problem described below:

There is an NxN grid.

<image>
(Figure when N = 3)

It costs 1 to move to an adjacent square. However, it cannot be moved diagonally. Also, there are M squares with obstacles, and you are not allowed to enter those squares. How many ways are there to move from mass (1, 1) to mass (N, N) at the lowest cost? The answer can be big, so take the mod at 1000000009 (= 109 + 9) and output it.

Constraints

* 2 ≤ N ≤ 106

* 0 ≤ M ≤ 50

* 1 ≤ Xi, Yi ≤ N

* If i ≠ j, then (Xi, Yi) ≠ (Xj, Yj)

* (Xi, Yi) ≠ (1, 1)

* (Xi, Yi) ≠ (N, N)

Input

Input is given in the following format

> N M
> X1 Y1
> X2 Y2
> ……
> XM YM
>

Xi, Yi means that there is an obstacle in (Xi, Yi)

Output

Output the total number of routes on one line. However, if there is no route to move from the mass (1, 1) to the mass (N, N), output 0.

Examples

Input

3 0


Output

6


Input

3 1
2 2


Output

2


Input

5 8
4 3
2 1
4 2
4 4
3 4
2 2
2 4
1 4


Output

1


Input

1000 10
104 87
637 366
393 681
215 604
707 876
943 414
95 327
93 415
663 596
661 842


Output

340340391


Input

2 2
1 2
2 1


Output

0

### Response

```cpp
#include <stdio.h>
#include <string.h>
#include <iostream>
#include <algorithm>
#include <utility>
#include <queue>
using namespace std;
#define rep(i, n) for (int i = 0; i < (int)(n); i++)
#define mp make_pair
#define fst first
#define snd second
typedef long long Int;
#define MOD (1000000009LL)
#define INF (1<<28)
inline void add(Int &a, Int b) { a = (a + b) % MOD; }

#define P (2000001)
Int inv[P], ft[P], ift[P];
const int dx[4] = {1, 0, -1, 0};
const int dy[4] = {0, 1, 0, -1};

int n, m;
pair<int, int> ps[64];
Int fw[64][64];
#define D (128)
#define H (52)
int xof[2][D][D], yof[2][D][D], dt[2][D][D];
Int cnt[2][D][D][2];

Int f(int sx, int sy, int ex, int ey) {
    const int dx = ex-sx, dy = ey-sy;
    return dx >= 0 && dy >= 0 ? ft[dx+dy] * ift[dx] % MOD * ift[dy] % MOD : 0;
}

Int calc(int sx, int sy, int ex, int ey) {
    static Int dp[64][2];
    rep (i, m) dp[i][0] = 0, dp[i][1] = f(sx, sy, ps[i].fst, ps[i].snd);
    Int ans = f(sx, sy, ex, ey);
    rep (i, m) rep (t, 2) {
        const Int w = dp[i][t] * f(ps[i].fst, ps[i].snd, ex, ey);
        if (t) add(ans, MOD-w%MOD); else add(ans, w);
        for (int j = i+1; j < m; j++) add(dp[j][(t+1)%2], dp[i][t]*fw[i][j]);
    }
    return ans;
}

inline bool valid(int w, int L, int x, int y) {
    return 0 <= x && x < L && 0 <= y && y < L && dt[w][x][y] < INF;
}

void build(int w, int L) {
    queue<pair<int, int> > q;
    q.push(mp(0, 0));
    memset(dt[w], -1, sizeof(dt[0]));
    dt[w][0][0] = 0;
    rep (x, L) rep (y, L) {
        if (w == 0) xof[w][x][y] = x, yof[w][x][y] = y;
        else xof[w][x][y] = n-1-x, yof[w][x][y] = n-1-y;
    }
    rep (x, L) rep (y, L) rep (i, m) {
        if (mp(xof[w][x][y], yof[w][x][y]) == ps[i]) dt[w][x][y] = INF;
    }
    while (!q.empty()) {
        const pair<int, int> vv = q.front();
        q.pop();
        const int x = vv.fst, y = vv.snd;
        rep (d, 4) {
            const int nx = x+dx[d], ny = y+dy[d];
            if (valid(w, L, nx, ny) && dt[w][nx][ny] == -1) {
                q.push(mp(nx, ny)), dt[w][nx][ny] = dt[w][x][y] + 1;
            }
        }
    }
    rep (x, L) rep (y, L) if (dt[w][x][y] == -1) dt[w][x][y] = INF;
    vector<pair<int, pair<int, int> > > vs;
    rep (x, L) rep (y, L) if (dt[w][x][y] < INF) {
        vs.push_back(mp(dt[w][x][y], mp(x, y)));
    }
    sort(vs.begin(), vs.end());
    memset(cnt[w], 0, sizeof(cnt[0]));
    cnt[w][0][0][1] = 1;
    rep (i, vs.size()) rep (t, 2) {
        const int x = vs[i].snd.fst, y = vs[i].snd.snd;
        rep (d, 4) {
            const int nx = x+dx[d], ny = y+dy[d];
            if (valid(w, L, nx, ny) && dt[w][nx][ny] == dt[w][x][y]+1) {
                add(cnt[w][nx][ny][d/2], cnt[w][x][y][t]);
            }
        }
    }
}

int main() {
    inv[1] = 1;
    for (int i = 2; i < P; i++) inv[i] = inv[MOD%i] * (MOD - MOD/i) % MOD;
    ft[0] = ift[0] = 1;
    rep (i, P-1) ft[i+1] = (i+1)*ft[i]%MOD, ift[i+1] = inv[i+1]*ift[i]%MOD;
    scanf("%d%d", &n, &m);
    rep (i, m) scanf("%d%d", &ps[i].fst, &ps[i].snd);
    rep (i, m) ps[i].fst--, ps[i].snd--;
    sort(ps, ps+m);
    rep (i, m) rep (j, m) {
        fw[i][j] = f(ps[i].fst, ps[i].snd, ps[j].fst, ps[j].snd);
    }
    if (n < D) {
        build(0, n);
        cout << cnt[0][n-1][n-1][0] << endl;
        return 0;
    }
    build(0, H), build(1, H);
    Int ans = 0;
    int cur = INF;
    vector<pair<int, int> > vs;
    rep (i, H) rep (j, H) if (dt[0][i][j] < INF && cnt[0][i][j][1]) {
        vs.push_back(mp(i, j));
    }
    rep (i, H) rep (j, H) if (dt[1][i][j] < INF && cnt[1][i][j][1]) {
        rep (k, vs.size()) {
            const int x = vs[k].fst, y = vs[k].snd;;
            const int sum = dt[0][x][y] + dt[1][i][j] 
                + xof[1][i][j] - x + yof[1][i][j] - y;
            if (sum > cur) continue;
            const Int w = calc(x, y, xof[1][i][j], yof[1][i][j]);
            if (w == 0) continue;
            if (cur > sum) cur = sum, ans = 0;
            if (cur == sum) add(ans, cnt[0][x][y][1]*w%MOD*cnt[1][i][j][1]);
        }
    }
    cout << ans << endl;
    return 0;
}
```