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:
#define  _CRT_SECURE_NO_WARNINGS
#define _USE_MATH_DEFINES

#pragma comment (linker, "/STACK:526000000")

#include "bits/stdc++.h"

using namespace std;
typedef string::const_iterator State;
#define eps 1e-5L
#define MAX_MOD 1000000007LL
#define GYAKU 500000004LL
#define seg_size 262144LL
#define MOD 998244353LL
#define pb push_back
#define mp make_pair
typedef long long ll;
#define REP(a,b) for(long long (a) = 0;(a) < (b);++(a))
#define ALL(x) (x).begin(),(x).end()

// geometry library

typedef complex<long double> Point;
typedef pair<complex<long double>, complex<long double>> Line;

typedef struct Circle {
    complex<long double> center;
    long double r;
}Circle;

//内積、 dot(a,b) = |a||b|cos()
long double dot(Point a, Point b) {
    return (a.real() * b.real() + a.imag() * b.imag());
}
//外積、cross(a,b) = |a||b|sin()
long double cross(Point a, Point b) {
    return (a.real() * b.imag() - a.imag() * b.real());
}

//線分と点の距離
long double Dist_Line_Point(Line a, Point b) {
    if (dot(a.second - a.first, b - a.first) < eps) return abs(b - a.first);
    if (dot(a.first - a.second, b - a.second) < eps) return abs(b - a.second);
    return abs(cross(a.second - a.first, b - a.first)) / abs(a.second - a.first);
}

//線分の交差判定
int is_intersected_ls(Line a, Line b) {
    return (cross(a.second - a.first, b.first - a.first) * cross(a.second - a.first, b.second - a.first) < 0) &&
        (cross(b.second - b.first, a.first - b.first) * cross(b.second - b.first, a.second - b.first) < 0);
}

//線分の交点
Point intersection_l(Line a, Line b) {
    Point da = a.second - a.first;
    Point db = b.second - b.first;
    return a.first + da * cross(db, b.first - a.first) / cross(db, da);
}

//線分と線分の距離
long double Dist_Line_Line(Line a, Line b) {
    if (is_intersected_ls(a, b) == 1) {
        return 0;
    }
    return min({ Dist_Line_Point(a,b.first), Dist_Line_Point(a,b.second),Dist_Line_Point(b,a.first),Dist_Line_Point(b,a.second) });
}

//円と円の交点
pair<Point, Point> intersection_Circle_Circle(Circle a, Circle b) {
    long double dist = abs(a.center - b.center);
    assert(dist <= eps + a.r + b.r);
    assert(dist+eps >= abs(a.r - b.r));
    Point target = b.center - a.center;
    long double pointer = target.real() * target.real() + target.imag() * target.imag();
    long double aa = pointer + a.r * a.r - b.r * b.r;
    aa /= 2.0L;
    Point l{ (aa * target.real() + target.imag() * sqrt(pointer * a.r * a.r - aa * aa))/pointer,
            (aa* target.imag() - target.real() * sqrt(pointer * a.r * a.r - aa * aa)) / pointer};
    Point r{ (aa * target.real() - target.imag() * sqrt(pointer * a.r * a.r - aa * aa)) / pointer,
        (aa * target.imag() + target.real() * sqrt(pointer * a.r * a.r - aa * aa)) / pointer };
    r = r + a.center;
    l = l + a.center;
    return mp(l, r);
}

//end of geometry
template<typename A>
A pows(A val, ll b) {
    assert(b >= 1);
    A ans = val;
    b--;
    while (b) {
        if (b % 2) {
            ans *= val;
        }
        val *= val;
        b /= 2LL;
    }
    return ans;
}

template<typename A>
class Matrix {
public:
    vector<vector<A>> data;
    Matrix(vector<vector<A>> a) :data(a){

    }
    Matrix operator + (const Matrix obj) {
        vector<vector<A>> ans;
        assert(obj.data.size() == this->data.size());
        assert(obj.data[0].size() == this->data[0].size());
        REP(i, obj.data.size()) {
            ans.push_back(vector<A>());
            REP(q, obj.data[i].size()) {
                A hoge = obj.data[i][q] + (this -> data[i][q]);
                ans.back().push_back(hoge);
            }
        }
        return Matrix(ans);
    }
    Matrix operator - (const Matrix obj) {
        vector<vector<A>> ans;
        assert(obj.data.size() == this->data.size());
        assert(obj.data[0].size() == this->data[0].size());
        REP(i, obj.data.size()) {
            ans.push_back(vector<A>());
            REP(q, obj.data[i].size()) {
                A hoge = this->data[i][q] - obj.data[i][q];
                ans.back().push_back(hoge);
            }
        }
        return Matrix(ans);
    }
    Matrix operator * (const Matrix obj) {
        vector<vector<A>> ans;
        assert(obj.data.size() == this->data[0].size());
        REP(i, this -> data.size()) {
            ans.push_back(vector<A>());
            REP(q, obj.data[0].size()) {
                A hoge = (this -> data[i][0]) * (obj.data[0][q]);
                for(int t = 1;t < obj.data[i].size();++t){
                    hoge += this -> data[i][t] * obj.data[t][q];
                }
                ans.back().push_back(hoge);
            }
        }
        return Matrix(ans);
    }
    Matrix &operator *= (const Matrix obj) {
        *this = (*this * obj);
        return *this;
    }
};

class modint {
public:
    using u64 = std::uint_fast64_t;
    u64 value = 0;
    u64 mod = 0;
    modint(ll a, ll b): value(a%b),mod(b) {

    }
    modint operator+(const modint rhs) const{
        return modint(*this) += rhs;
    }
    modint operator-(const modint rhs) const{
        return modint(*this) -= rhs;
    }
    modint operator*(const modint rhs) const {
        return modint(*this) *= rhs;
    }
    modint operator/(const modint rhs) const{
        return modint(*this) /= rhs;
    }
    modint& operator+=(const modint rhs) {
        assert(rhs.mod == mod);
        value += rhs.value;
        if (value >= mod) {
            value -= mod;
        }
        return *this;
    }
    modint& operator-=(const modint rhs) {
        assert(rhs.mod == mod);
        if (value < rhs.value) {
            value += mod;
        }
        value -= rhs.value;
        return *this;
    }
    modint& operator*=(const modint rhs) {
        assert(rhs.mod == mod);
        value = (value *  rhs.value) % (ll)(1e9+9);
        return *this;
    }
    modint& operator/=(modint rhs) {
        assert(rhs.mod == mod);
        ll rem = mod - 2;
        while (rem) {
            if (rem % 2) {
                *this *= rhs;
            }
            rhs *= rhs;
            rem /= 2LL;
        }
        return *this;
    }
};

void init() {
    iostream::sync_with_stdio(false);
    cout << fixed << setprecision(20);
}

unsigned long xor128() {
    static unsigned long x = 123456789, y = 362436069, z = 521288629, w = 88675123;
    unsigned long t = (x ^ (x << 11));
    x = y; y = z; z = w;
    return (w = (w ^ (w >> 19)) ^ (t ^ (t >> 8)));
}

#define int ll
void solve(){
    const int mod = 1000000009LL;
    REP(itea,100000){
        int w, h, n;
        cin >> w >> h >> n;
        if (w == 0) return;
        cout << "Case " << itea + 1 << ": ";
        map<int, vector<int>> inputs;
        inputs[h].push_back(-1);
        REP(i, n) {
            int a, b;
            cin >> a >> b;
            a--;
            inputs[b].push_back(a);
        }
        int back = 1;
        vector<modint> ans;
        ans.push_back(modint{ 1,mod });
        REP(i, w-1) {
            ans.push_back(modint{ 0 ,mod });
        }
        Matrix<modint> base(vector<vector<modint>>{ans});
        vector<vector<modint>> multier;
        REP(i, w) {
            multier.push_back(vector<modint>());
            REP(q, w) {
                if (i - 1 <= q && q <= i + 1) {
                    multier.back().push_back(modint(1, mod));
                }
                else {
                    multier.back().push_back(modint(0, mod));
                }
            }
        }
        Matrix<modint> multi(multier);
        for (auto x : inputs) {
            if(x.first - back > 0){
                base *= pows<Matrix<modint>>(multi, x.first - back);
                back = x.first;
            }
            for (auto y : x.second) {
                if (y >= 0) {
                    base.data[0][y] = modint(0, mod);
                }
            }
        }
        cout << base.data[0].back().value << endl;
    }
}

#undef int
int main() {
    init();
    solve();
}