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<stdio.h>
#include <iostream>
#include <math.h>
#include <numeric>
#include <vector>
#include <map>
#include <functional>
#include <stdio.h>
#include <array>
#include <algorithm>
#include <string>
#include <assert.h>
#include <stdio.h>
#include <queue>
#include<iomanip>
#include<bitset>
#include<stack>
#include<set>
using namespace std;

#include<assert.h>
const int mod = 1000000009;

struct Mod {
	int num;
	Mod() : num(0) { ; }
	Mod(long long int n) : num((n % mod + mod) % mod) { ; }
	Mod(int n) : num((n % mod + mod) % mod) { ; }
	operator int() { return num; }
};

Mod operator+(Mod a, Mod b) { return Mod((a.num + b.num) % mod); }
Mod operator+(long long int a, Mod b) { return Mod(a) + b; }
Mod operator-(Mod a, Mod b) { return Mod((mod + a.num - b.num) % mod); }
Mod operator-(long long int a, Mod b) { return Mod(a) - b; }
Mod operator*(Mod a, Mod b) { return Mod(((long long)a.num * b.num) % mod); }
Mod operator*(long long int a, Mod b) { return Mod(a)*b; }
Mod operator+=(Mod &a, Mod b) { return a = a + b; }
Mod operator+=(long long int &a, Mod b) { return a = a + b; }
Mod operator-=(Mod &a, Mod b) { return a = a - b; }
Mod operator-=(long long int &a, Mod b) { return a = a - b; }
Mod operator*=(Mod &a, Mod b) { return a = a * b; }
Mod operator*=(long long int &a, Mod b) { return a = a * b; }
Mod operator^(Mod a, int n) {
	if (n == 0) return Mod(1);
	Mod res = (a * a) ^ (n / 2);
	if (n % 2) res = res * a;
	return res;
}
Mod inv(Mod a) { return a ^ (mod - 2); }
Mod operator/(Mod a, Mod b) {
	assert(b != 0);
	return a * inv(b);
}
Mod operator/(long long int a, Mod b) {
	assert(b != 0);
	return Mod(a) * inv(b);
}

#define MAX_N 1024000

Mod fact[MAX_N], factinv[MAX_N];
void init() {
	fact[0] = Mod(1); factinv[0] = 1;
	for (int i = 0; i < MAX_N - 1; ++i) {
		fact[i + 1] = fact[i] * Mod(i + 1);
		factinv[i + 1] = factinv[i] / Mod(i + 1);
	}
}
Mod comb(int a, int b) {
	return fact[a] * factinv[b] * factinv[a - b];
}


vector<vector<Mod>> keisann(const vector<vector<Mod>>l, const vector<vector<Mod>>r) {
	vector<vector<long long int>> ll(l.size(),vector<long long int>(l[0].size())), rr(r.size(), vector<long long int>(r[0].size()));
	for (int i = 0; i < l.size(); ++i) {
		for (int j = 0; j < l[0].size(); ++j){
			ll[i][j] = l[i][j].num;
		}
	}
	for (int i = 0; i < r.size(); ++i) {
		for (int j = 0; j < r[0].size(); ++j) {
			rr[i][j] = r[i][j].num;
		}
	}
	vector<vector<Mod>>ans(l.size(), vector<Mod>(r[0].size()));
	assert(l[0].size() == r.size());
	for (int h = 0; h < l.size(); ++h) {
		for (int w = 0; w < r[0].size(); ++w) {
			long long int aa=0;
			for (int i = 0; i < r.size(); ++i) {

				aa+= (ll[h][i] * rr[i][w])%mod;
				//ans[h][w] += l[h][i] * r[i][w];
			}
			ans[h][w] = aa%mod;
		}
	}
	return ans;
}

vector<vector<Mod>>powgyou(vector<vector<Mod>>a, const long long int n) {
	assert(a.size() == a[0].size());
	if (n == 0) {
		vector<vector<Mod>>ans(a);
		for (int i = 0; i < ans.size(); ++i) {
			for (int j = 0; j < ans.size(); ++j) {
				ans[i][j] = 0;
			}
		}
		for (int i = 0; i < ans.size(); ++i) {
			ans[i][i] = 1;
		}
		return ans;
	}
	if (n == 1)return a;
	else {
		vector<vector<Mod>>ans(a.size(), vector<Mod>(a[0].size(), 0));
		ans = powgyou(a, n / 2);
		ans = keisann(ans, ans);
		if (n % 2) {
			ans = keisann(ans, a);
		}
		return ans;
		
	}
}
//#define FOR(i,a,b) for(int i=(a);i<(b);++i)
//#define RFOR(i,a,b) for(int i=(a);i>=(b);--i)
//class segtree
//{
//public:
//	vector<int> seg;
//	int size;
//	segtree(int n)
//	{
//		size = 1;
//		while (size < n)  size *= 2;
//		seg.resize(size * 2, -1);
//	}
//
//	void update(int k, int v)
//	{
//		k += size - 1;
//		seg[k] = v;
//		while (k > 0)
//		{
//			k = (k - 1) / 2;
//			seg[k] = min(seg[k * 2 + 1], seg[k * 2 + 2]);
//		}
//	}
//
//	int query(int v)
//	{
//		int k = 0;
//		while (k < size - 1)
//		{
//			if (seg[k * 2 + 1] < v)
//			{
//				k = k * 2 + 1;
//			}
//			else
//			{
//				k = k * 2 + 2;
//			}
//		}
//		return k - (size - 1);
//	}
//};


int main()
{
	int i = 0;
	while (1) {
		i++;
		long long int W, H, N; cin >> W >> H >> N;
		if (!W)break;
		map<long long int, vector<int >>bs;
		for (int i = 0; i < N; ++i) {
			long long int x, y; cin >> x >> y; x--; y--;
			bs[y].push_back(x);
		}

		long long int pre = 0;
		vector<vector<Mod>>gyou(W, vector<Mod>(1, 0));
		gyou[0][0] = 1;

		vector<vector<Mod>>Kakeru(W, vector<Mod>(W));
		for (int i = 0; i < W; ++i) {
			for (int j = 0; j < W; ++j) {
				if (abs(i - j) <= 1) {
					Kakeru[i][j] = Mod(1);
				}
			}
		}

		for (auto it = bs.begin(); it != bs.end(); ++it) {
			if (it->first == 0) {
				continue;
			}
			else {
				vector<vector<Mod>>a(powgyou(Kakeru, (*it).first - pre));
				vector<vector<Mod>>b(keisann(a, gyou));
				vector<int>c((*it).second);
				for (int j = 0; j < c.size(); ++j) {
					b[c[j]][0] = 0;
				}
				gyou = b;
				pre = (*it).first;
			}
		}

		vector<vector<Mod>>a(powgyou(Kakeru, H-1 - pre));
		vector<vector<Mod>>b(keisann(a, gyou));

		Mod ans = b[W - 1][0];
		cout << "Case " << i << ": " << ans << endl;
	}
	
	return 0;
	
}