Problem: There are n cells, numbered 1,2,..., n from left to right. You have to place a robot at any cell initially. The robot must make exactly k moves.

In one move, the robot must move one cell to the left or right, provided that it doesn't move out of bounds. In other words, if the robot was in the cell i, it must move to either the cell i-1 or the cell i+1, as long as it lies between 1 and n (endpoints inclusive). The cells, in the order they are visited (including the cell the robot is placed), together make a good path.

Each cell i has a value a_i associated with it. Let c_0, c_1, ..., c_k be the sequence of cells in a good path in the order they are visited (c_0 is the cell robot is initially placed, c_1 is the cell where the robot is after its first move, and so on; more formally, c_i is the cell that the robot is at after i moves). Then the value of the path is calculated as a_{c_0} + a_{c_1} + ... + a_{c_k}.

Your task is to calculate the sum of values over all possible good paths. Since this number can be very large, output it modulo 10^9 + 7. Two good paths are considered different if the starting cell differs or there exists an integer i ∈ [1, k] such that the current cell of the robot after exactly i moves is different in those paths.

You must process q updates to a and print the updated sum each time. Each update changes the value of exactly one cell. See the input format and the sample input-output for more details.

Input

The first line of the input contains three space-separated integers n, k and q (2 ≤ n ≤ 5000; 1 ≤ k ≤ 5000; 1 ≤ q ≤ 2 ⋅ 10^5).

The second line of the input contains n integers a_1, a_2, ..., a_n (1 ≤ a_i ≤ 10^9).

q lines follow. Each line contains two space-separated integers i and x (1 ≤ i ≤ n; 1 ≤ x ≤ 10^9) indicating that you must change the value of a_i to x.

Output

Print q integers. The i-th integer should be the sum of values over all good paths after the first i updates are performed. Since the answers may be large, print them modulo 10^9 + 7.

Examples

Input


5 1 5
3 5 1 4 2
1 9
2 4
3 6
4 6
5 2


Output


62
58
78
86
86


Input


5 2 5
3 5 1 4 2
1 9
2 4
3 6
4 6
5 2


Output


157
147
207
227
227


Input


4 40 6
92 21 82 46
3 56
1 72
4 28
1 97
2 49
2 88


Output


239185261
666314041
50729936
516818968
766409450
756910476

Note

In the first example, the good paths are (1, 2), (2, 1), (2, 3), (3, 2), (3, 4), (4, 3), (4, 5), (5, 4).

Initially the values of a are [3, 5, 1, 4, 2]. After the first update, they become [9, 5, 1, 4, 2]. After the second update, they become [9, 4, 1, 4, 2], and so on.

Solution (in CPP):

Solution:
// Jai Shree Ram  
  
#include<bits/stdc++.h>
using namespace std;

#define rep(i,a,n)     for(int i=a;i<n;i++)
#define ll             long long
#define int            long long
#define pb             push_back
#define all(v)         v.begin(),v.end()
#define endl           "\n"
#define x              first
#define y              second
#define gcd(a,b)       __gcd(a,b)
#define mem1(a)        memset(a,-1,sizeof(a))
#define mem0(a)        memset(a,0,sizeof(a))
#define sz(a)          (int)a.size()
#define pii            pair<int,int>
#define hell           1000000007
#define elasped_time   1.0 * clock() / CLOCKS_PER_SEC



template<typename T1,typename T2>istream& operator>>(istream& in,pair<T1,T2> &a){in>>a.x>>a.y;return in;}
template<typename T1,typename T2>ostream& operator<<(ostream& out,pair<T1,T2> a){out<<a.x<<" "<<a.y;return out;}
template<typename T,typename T1>T maxs(T &a,T1 b){if(b>a)a=b;return a;}
template<typename T,typename T1>T mins(T &a,T1 b){if(b<a)a=b;return a;}

const int MOD = hell;
 
struct mod_int {
    int val;
 
    mod_int(long long v = 0) {
        if (v < 0)
            v = v % MOD + MOD;
 
        if (v >= MOD)
            v %= MOD;
 
        val = v;
    }
 
    static int mod_inv(int a, int m = MOD) {
        int g = m, r = a, x = 0, y = 1;
 
        while (r != 0) {
            int q = g / r;
            g %= r; swap(g, r);
            x -= q * y; swap(x, y);
        }
 
        return x < 0 ? x + m : x;
    }
 
    explicit operator int() const {
        return val;
    }
 
    mod_int& operator+=(const mod_int &other) {
        val += other.val;
        if (val >= MOD) val -= MOD;
        return *this;
    }
 
    mod_int& operator-=(const mod_int &other) {
        val -= other.val;
        if (val < 0) val += MOD;
        return *this;
    }
 
    static unsigned fast_mod(uint64_t x, unsigned m = MOD) {
           #if !defined(_WIN32) || defined(_WIN64)
                return x % m;
           #endif
           unsigned x_high = x >> 32, x_low = (unsigned) x;
           unsigned quot, rem;
           asm("divl %4\n"
            : "=a" (quot), "=d" (rem)
            : "d" (x_high), "a" (x_low), "r" (m));
           return rem;
    }
 
    mod_int& operator*=(const mod_int &other) {
        val = fast_mod((uint64_t) val * other.val);
        return *this;
    }
 
    mod_int& operator/=(const mod_int &other) {
        return *this *= other.inv();
    }
 
    friend mod_int operator+(const mod_int &a, const mod_int &b) { return mod_int(a) += b; }
    friend mod_int operator-(const mod_int &a, const mod_int &b) { return mod_int(a) -= b; }
    friend mod_int operator*(const mod_int &a, const mod_int &b) { return mod_int(a) *= b; }
    friend mod_int operator/(const mod_int &a, const mod_int &b) { return mod_int(a) /= b; }
 
    mod_int& operator++() {
        val = val == MOD - 1 ? 0 : val + 1;
        return *this;
    }
 
    mod_int& operator--() {
        val = val == 0 ? MOD - 1 : val - 1;
        return *this;
    }
 
    mod_int operator++(int32_t) { mod_int before = *this; ++*this; return before; }
    mod_int operator--(int32_t) { mod_int before = *this; --*this; return before; }
 
    mod_int operator-() const {
        return val == 0 ? 0 : MOD - val;
    }
 
    bool operator==(const mod_int &other) const { return val == other.val; }
    bool operator!=(const mod_int &other) const { return val != other.val; }
 
    mod_int inv() const {
        return mod_inv(val);
    }
 
    mod_int pow(long long p) const {
        assert(p >= 0);
        mod_int a = *this, result = 1;
 
        while (p > 0) {
            if (p & 1)
                result *= a;
 
            a *= a;
            p >>= 1;
        }
 
        return result;
    }
 
    friend ostream& operator<<(ostream &stream, const mod_int &m) {
        return stream << m.val;
    }
    friend istream& operator >> (istream &stream, mod_int &m) {
        return stream>>m.val;   
    }
};

int solve(){
 		int n,k,q;cin>>n>>k>>q;
 		vector<int>a(n+1);
 		rep(i,1,n+1){
 			cin>>a[i];
 		}
 		vector<vector<mod_int>>dp(k+1,vector<mod_int>(n+2));
 		vector<mod_int>co(n+1);
 		rep(i,1,n+1){
 			dp[0][i]=1;
 		}
 		rep(i,1,k+1){
 			rep(j,1,n+1){
 				dp[i][j]=dp[i-1][j+1]+dp[i-1][j-1];
 			}
 		}
 		mod_int ans=0;
 		rep(i,1,n+1){
 			rep(j,0,k+1){
 				co[i]+=dp[j][i]*dp[k-j][i];
 			}
 			ans+=a[i]*co[i];
 		}
 		rep(i,0,q){
 			int j,x;cin>>j>>x;
 			ans-=a[j]*co[j];a[j]=x;
 			ans+=a[j]*co[j];
 			cout<<ans<<endl;
 		}
 return 0;	
}
signed main(){
    ios_base::sync_with_stdio(0);cin.tie(0);cout.tie(0);
    //freopen("input.txt", "r", stdin);
    //freopen("output.txt", "w", stdout);
    #ifdef SIEVE
    sieve();
    #endif
    #ifdef NCR
    init();
    #endif
    int t=1;//cin>>t;
    while(t--){
        solve();
    }
    return 0;
}