Problem: There is a grid that consists of W \times H cells. The upper-left-most cell is (1, 1). You are standing on the cell of (1,1) and you are going to move to cell of (W, H). You can only move to adjacent lower-left, lower or lower-right cells.

There are obstructions on several cells. You can not move to it. You cannot move out the grid, either. Write a program that outputs the number of ways to reach (W,H) modulo 1,000,000,009. You can assume that there is no obstruction at (1,1).



Input

The first line contains three integers, the width W, the height H, and the number of obstructions N. (1 \leq W \leq 75, 2 \leq H \leq 10^{18}, 0 \leq N \leq 30) Each of following N lines contains 2 integers, denoting the position of an obstruction (x_i, y_i).

The last test case is followed by a line containing three zeros.

Output

For each test case, print its case number and the number of ways to reach (W,H) modulo 1,000,000,009.

Example

Input

2 4 1
2 1
2 2 1
2 2
0 0 0


Output

Case 1: 4
Case 2: 0

Solution (in CPP):

Solution:
#include <vector>
#include <iostream>
#include <algorithm>
#pragma warning(disable : 4996)
using namespace std;
class ModMatrix {
public:
	// ------ Variables ------ //
	int r, c; vector<vector<int> > mat;
	static const int mod = 1000000009;
	// ------ Constructor ------ //
	ModMatrix() : r(0), c(0) {}
	ModMatrix(int r_, int c_) : r(r_), c(c_) { mat = vector<vector<int> >(r_, vector<int>(c_, 0)); }
	// ------ Operator ------ //
	friend bool operator==(const ModMatrix& a, const ModMatrix& b) { return a.mat == b.mat; }
	friend bool operator!=(const ModMatrix& a, const ModMatrix& b) { return a.mat != b.mat; }
	friend ModMatrix operator+(const ModMatrix& a, const ModMatrix& b) {
		if (a.r == b.r && a.c == b.c) {
			ModMatrix c(a);
			for (int i = 0; i < a.r; i++) {
				for (int j = 0; j < a.c; j++) {
					c.mat[i][j] = (c.mat[i][j] + b.mat[i][j]) % ModMatrix::mod;
				}
			}
			return c;
		}
		return ModMatrix();
	}
	friend ModMatrix operator-(const ModMatrix& a, const ModMatrix& b) {
		if (a.r == b.r && a.c == b.c) {
			ModMatrix c(a);
			for (int i = 0; i < a.r; i++) {
				for (int j = 0; j < a.c; j++) {
					c.mat[i][j] = (c.mat[i][j] - b.mat[i][j] + ModMatrix::mod) % ModMatrix::mod;
				}
			}
			return c;
		}
		return ModMatrix();
	}
	friend ModMatrix operator*(const ModMatrix& a, const ModMatrix& b) {
		if (a.c == b.r) {
			ModMatrix c(a.r, b.c);
			for (int i = 0; i < a.r; i++) {
				for (int j = 0; j < b.c; j++) {
					for (int k = 0; k < a.c; k++) {
						c.mat[i][j] = (c.mat[i][j] + 1LL * a.mat[i][k] * b.mat[k][j]) % ModMatrix::mod;
					}
				}
			}
			return c;
		}
		return ModMatrix();
	}
	template<class Type>
	friend ModMatrix operator*(const ModMatrix& a, const Type& b) {
		ModMatrix c(a.r, a.c);
		for (int i = 0; i < a.r; i++) {
			for (int j = 0; j < a.c; j++) {
				c.mat[i][j] = 1LL * c.mat[i][j] * b % ModMatrix::mod;
			}
		}
		return c;
	}
	template<class Type>
	friend ModMatrix operator*(const Type& a, const ModMatrix& b) {
		ModMatrix c(b.r, b.c);
		for (int i = 0; i < b.r; i++) {
			for (int j = 0; j < b.c; j++) {
				c.mat[i][j] = 1LL * c.mat[i][j] * a % ModMatrix::mod;
			}
		}
		return c;
	}
	ModMatrix& operator+=(const ModMatrix& a) {
		if (a.r == r && a.c == c) {
			for (int i = 0; i < r; i++) {
				for (int j = 0; j < c; j++) {
					mat[i][j] = (mat[i][j] + a.mat[i][j]) % mod;
				}
			}
		}
		return *this;
	}
	ModMatrix& operator-=(const ModMatrix& a) {
		if (a.r == r && a.c == c) {
			for (int i = 0; i < r; i++) {
				for (int j = 0; j < c; j++) {
					mat[i][j] = (mat[i][j] - a.mat[i][j] + mod) % mod;
				}
			}
		}
		return *this;
	}
	ModMatrix& operator*=(const ModMatrix& a) {
		if (c == a.r && c == a.c) {
			vector<vector<int> > tmp(r, vector<int>(c));
			for (int i = 0; i < r; i++) {
				for (int j = 0; j < c; j++) {
					for (int k = 0; k < c; k++) {
						tmp[i][j] = (tmp[i][j] + 1LL * mat[i][k] * a.mat[k][j]) % mod;
					}
				}
			}
			for (int i = 0; i < r; i++) {
				for (int j = 0; j < c; j++) {
					mat[i][j] = tmp[i][j];
				}
			}
		}
		return *this;
	}
	template<class Type>ModMatrix& operator*=(const Type& a) {
		for (int i = 0; i < r; i++) {
			for (int j = 0; j < c; j++) {
				mat[i][j] = 1LL * mat[i][j] * c % mod;
			}
		}
		return *this;
	}
};
// ------ Function ------ //
ModMatrix unit(int r) {
	ModMatrix res(r, r);
	for (int i = 0; i < r; i++) {
		res.mat[i][i] = 1;
	}
	return res;
}
ModMatrix power(ModMatrix a, long long b) {
	if (a.r == a.c) {
		ModMatrix res = unit(a.r);
		for (int i = 62; i >= 0; i--) {
			res *= res;
			if (b & (1LL << i)) {
				res *= a;
			}
		}
		return res;
	}
	return ModMatrix();
}
int W, N, C, A[30]; long long H, B[30], R[30], pos, nxt; vector<int> S[30];
int main() {
	while (true) {
		scanf("%d%lld%d", &W, &H, &N);
		if (W == 0 && H == 0 && N == 0) break;
		for (int i = 0; i < N; i++) scanf("%d%lld", &A[i], &B[i]);
		ModMatrix F(W, W), G(W, 1);
		for (int i = 0; i < W; i++) {
			for (int j = 0; j < W; j++) {
				if (-1 <= j - i && j - i <= 1) {
					F.mat[i][j] = 1;
				}
			}
		}
		int c = 0;
		for (int i = 0; i < N; i++) {
			bool flag = true;
			for (int j = 0; j < c; j++) {
				if (R[j] == B[i]) flag = false;
			}
			if (flag) R[c++] = B[i];
		}
		for (int i = 0; i < c; i++) S[i].clear();
		sort(R, R + c);
		for (int i = 0; i < N; i++) {
			for (int j = 0; j < c; j++) {
				if (R[j] == B[i]) {
					S[j].push_back(A[i]);
				}
			}
		}
		G.mat[0][0] = 1;
		pos = 1;
		if (N == 0) nxt = H;
		else nxt = R[0];
		for (int i = 0; i <= c; i++) {
			G = power(F, nxt - pos) * G;
			if (i != c) {
				for (int j = 0; j < S[i].size(); j++) {
					G.mat[S[i][j] - 1][0] = 0;
				}
			}
			pos = nxt;
			if (i + 1 == c) nxt = H;
			else nxt = R[i + 1];
		}
		printf("Case %d: %d\n", ++C, G.mat[W - 1][0]);
	}
	return 0;
}