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 JAVA):

Solution:
import java.io.*;
import java.util.*;

import static java.lang.Math.*;
import static java.util.Arrays.*;

public class cf1467d {

    public static void main(String[] args) throws IOException {
        int n = rni(), k = ni(), q = ni(), a[] = ria(n), dp[][] = new int[k + 1][n], sz[][] = new int[k + 1][n], mul[] = new int[n];
        fill(dp[0], 1);
        for (int i = 0; i < k; ++i) {
            dp[i + 1][0] = dp[i][1];
            dp[i + 1][n - 1] = dp[i][n - 2];
            for (int j = 1; j < n - 1; ++j) {
                dp[i + 1][j] = madd(dp[i][j - 1], dp[i][j + 1]);
            }
        }
        fill(sz[k], 1);
        for (int i = k - 1; i >= 0; --i) {
            sz[i][0] = sz[i + 1][1];
            sz[i][n - 1] = sz[i + 1][n - 2];
            for (int j = 1; j < n - 1; ++j) {
                sz[i][j] = madd(sz[i + 1][j - 1], sz[i + 1][j + 1]);
            }
        }
        for (int i = 0; i <= k; ++i) {
            for (int j = 0; j < n; ++j) {
                mul[j] = madd(mul[j], mmul(dp[i][j], sz[i][j]));
            }
        }
        int ans = 0;
        for (int i = 0; i < n; ++i) {
            ans = madd(ans, mmul(mul[i], a[i]));
        }
        while (q --> 0) {
            int i = rni() - 1, x = ni();
            ans = msub(ans, mmul(mul[i], a[i]));
            a[i] = x;
            ans = madd(ans, mmul(mul[i], a[i]));
            prln(ans);
        }
        close();
    }

    static int mmod = 1000000007;

    static int madd(int a, int b) {
        return (a + b) % mmod;
    }

    static int madd(int... a) {
        int ans = a[0];
        for (int i = 1; i < a.length; ++i) {
            ans = madd(ans, a[i]);
        }
        return ans;
    }

    static int msub(int a, int b) {
        return (a - b + mmod) % mmod;
    }

    static int mmul(int a, int b) {
        return (int) ((long) a * b % mmod);
    }

    static int mmul(int... a) {
        int ans = a[0];
        for (int i = 1; i < a.length; ++i) {
            ans = mmul(ans, a[i]);
        }
        return ans;
    }

    static int minv(int x) {
        // return mpow(x, mmod - 2);
        return (exgcd(x, mmod)[0] % mmod + mmod) % mmod;
    }

    static int mpow(int a, long b) {
        if (a == 0) {
            return 0;
        }
        int ans = 1;
        while (b > 0) {
            if ((b & 1) > 0) {
                ans = mmul(ans, a);
            }
            a = mmul(a, a);
            b >>= 1;
        }
        return ans;
    }

    static BufferedReader __i = new BufferedReader(new InputStreamReader(System.in));
    static PrintWriter __o = new PrintWriter(new OutputStreamWriter(System.out));
    static StringTokenizer input;
    static Random __r = new Random();

    // references
    // IBIG = 1e9 + 7
    // IMAX ~= 2e9
    // LMAX ~= 9e18

    // constants
    static final int IBIG = 1000000007;
    static final int IMAX = 2147483647;
    static final long LMAX = 9223372036854775807L;
    // math util
    static int minof(int a, int b, int c) {return min(a, min(b, c));}
    static int minof(int... x) {if (x.length == 1) return x[0]; if (x.length == 2) return min(x[0], x[1]); if (x.length == 3) return min(x[0], min(x[1], x[2])); int min = x[0]; for (int i = 1; i < x.length; ++i) if (x[i] < min) min = x[i]; return min;}
    static long minof(long a, long b, long c) {return min(a, min(b, c));}
    static long minof(long... x) {if (x.length == 1) return x[0]; if (x.length == 2) return min(x[0], x[1]); if (x.length == 3) return min(x[0], min(x[1], x[2])); long min = x[0]; for (int i = 1; i < x.length; ++i) if (x[i] < min) min = x[i]; return min;}
    static int maxof(int a, int b, int c) {return max(a, max(b, c));}
    static int maxof(int... x) {if (x.length == 1) return x[0]; if (x.length == 2) return max(x[0], x[1]); if (x.length == 3) return max(x[0], max(x[1], x[2])); int max = x[0]; for (int i = 1; i < x.length; ++i) if (x[i] > max) max = x[i]; return max;}
    static long maxof(long a, long b, long c) {return max(a, max(b, c));}
    static long maxof(long... x) {if (x.length == 1) return x[0]; if (x.length == 2) return max(x[0], x[1]); if (x.length == 3) return max(x[0], max(x[1], x[2])); long max = x[0]; for (int i = 1; i < x.length; ++i) if (x[i] > max) max = x[i]; return max;}
    static int powi(int a, int b) {if (a == 0) return 0; int ans = 1; while (b > 0) {if ((b & 1) > 0) ans *= a; a *= a; b >>= 1;} return ans;}
    static long powl(long a, int b) {if (a == 0) return 0; long ans = 1; while (b > 0) {if ((b & 1) > 0) ans *= a; a *= a; b >>= 1;} return ans;}
    static int fli(double d) {return (int) d;}
    static int cei(double d) {return (int) ceil(d);}
    static long fll(double d) {return (long) d;}
    static long cel(double d) {return (long) ceil(d);}
    static int gcf(int a, int b) {return b == 0 ? a : gcf(b, a % b);}
    static long gcf(long a, long b) {return b == 0 ? a : gcf(b, a % b);}
    static int[] exgcd(int a, int b) {if (b == 0) return new int[] {1, 0}; int[] y = exgcd(b, a % b); return new int[] {y[1], y[0] - y[1] * (a / b)};}
    static long[] exgcd(long a, long b) {if (b == 0) return new long[] {1, 0}; long[] y = exgcd(b, a % b); return new long[] {y[1], y[0] - y[1] * (a / b)};}
    static int randInt(int min, int max) {return __r.nextInt(max - min + 1) + min;}
    static long mix(long x) {x += 0x9e3779b97f4a7c15L; x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9L; x = (x ^ (x >> 27)) * 0x94d049bb133111ebL; return x ^ (x >> 31);}
    // array util
    static void reverse(int[] a) {for (int i = 0, n = a.length, half = n / 2; i < half; ++i) {int swap = a[i]; a[i] = a[n - i - 1]; a[n - i - 1] = swap;}}
    static void reverse(long[] a) {for (int i = 0, n = a.length, half = n / 2; i < half; ++i) {long swap = a[i]; a[i] = a[n - i - 1]; a[n - i - 1] = swap;}}
    static void reverse(double[] a) {for (int i = 0, n = a.length, half = n / 2; i < half; ++i) {double swap = a[i]; a[i] = a[n - i - 1]; a[n - i - 1] = swap;}}
    static void reverse(char[] a) {for (int i = 0, n = a.length, half = n / 2; i < half; ++i) {char swap = a[i]; a[i] = a[n - i - 1]; a[n - i - 1] = swap;}}
    static void shuffle(int[] a) {int n = a.length - 1; for (int i = 0; i < n; ++i) {int ind = randInt(i, n); int swap = a[i]; a[i] = a[ind]; a[ind] = swap;}}
    static void shuffle(long[] a) {int n = a.length - 1; for (int i = 0; i < n; ++i) {int ind = randInt(i, n); long swap = a[i]; a[i] = a[ind]; a[ind] = swap;}}
    static void shuffle(double[] a) {int n = a.length - 1; for (int i = 0; i < n; ++i) {int ind = randInt(i, n); double swap = a[i]; a[i] = a[ind]; a[ind] = swap;}}
    static void rsort(int[] a) {shuffle(a); sort(a);}
    static void rsort(long[] a) {shuffle(a); sort(a);}
    static void rsort(double[] a) {shuffle(a); sort(a);}
    static int[] copy(int[] a) {int[] ans = new int[a.length]; for (int i = 0; i < a.length; ++i) ans[i] = a[i]; return ans;}
    static long[] copy(long[] a) {long[] ans = new long[a.length]; for (int i = 0; i < a.length; ++i) ans[i] = a[i]; return ans;}
    static double[] copy(double[] a) {double[] ans = new double[a.length]; for (int i = 0; i < a.length; ++i) ans[i] = a[i]; return ans;}
    static char[] copy(char[] a) {char[] ans = new char[a.length]; for (int i = 0; i < a.length; ++i) ans[i] = a[i]; return ans;}
    // input
    static void r() throws IOException {input = new StringTokenizer(rline());}
    static int ri() throws IOException {return Integer.parseInt(rline());}
    static long rl() throws IOException {return Long.parseLong(rline());}
    static double rd() throws IOException {return Double.parseDouble(rline());}
    static int[] ria(int n) throws IOException {int[] a = new int[n]; r(); for (int i = 0; i < n; ++i) a[i] = ni(); return a;}
    static int[] riam1(int n) throws IOException {int[] a = new int[n]; r(); for (int i = 0; i < n; ++i) a[i] = ni() - 1; return a;}
    static long[] rla(int n) throws IOException {long[] a = new long[n]; r(); for (int i = 0; i < n; ++i) a[i] = nl(); return a;}
    static double[] rda(int n) throws IOException {double[] a = new double[n]; r(); for (int i = 0; i < n; ++i) a[i] = nd(); return a;}
    static char[] rcha() throws IOException {return rline().toCharArray();}
    static String rline() throws IOException {return __i.readLine();}
    static String n() {return input.nextToken();}
    static int rni() throws IOException {r(); return ni();}
    static int ni() {return Integer.parseInt(n());}
    static long rnl() throws IOException {r(); return nl();}
    static long nl() {return Long.parseLong(n());}
    static double rnd() throws IOException {r(); return nd();}
    static double nd() {return Double.parseDouble(n());}
    // output
    static void pr(int i) {__o.print(i);}
    static void prln(int i) {__o.println(i);}
    static void pr(long l) {__o.print(l);}
    static void prln(long l) {__o.println(l);}
    static void pr(double d) {__o.print(d);}
    static void prln(double d) {__o.println(d);}
    static void pr(char c) {__o.print(c);}
    static void prln(char c) {__o.println(c);}
    static void pr(char[] s) {__o.print(new String(s));}
    static void prln(char[] s) {__o.println(new String(s));}
    static void pr(String s) {__o.print(s);}
    static void prln(String s) {__o.println(s);}
    static void pr(Object o) {__o.print(o);}
    static void prln(Object o) {__o.println(o);}
    static void prln() {__o.println();}
    static void pryes() {prln("yes");}
    static void pry() {prln("Yes");}
    static void prY() {prln("YES");}
    static void prno() {prln("no");}
    static void prn() {prln("No");}
    static void prN() {prln("NO");}
    static boolean pryesno(boolean b) {prln(b ? "yes" : "no"); return b;};
    static boolean pryn(boolean b) {prln(b ? "Yes" : "No"); return b;}
    static boolean prYN(boolean b) {prln(b ? "YES" : "NO"); return b;}
    static void prln(int... a) {for (int i = 0, len = a.length - 1; i < len; pr(a[i]), pr(' '), ++i); if (a.length > 0) prln(a[a.length - 1]); else prln();}
    static void prln(long... a) {for (int i = 0, len = a.length - 1; i < len; pr(a[i]), pr(' '), ++i); if (a.length > 0) prln(a[a.length - 1]); else prln();}
    static void prln(double... a) {for (int i = 0, len = a.length - 1; i < len; pr(a[i]), pr(' '), ++i); if (a.length > 0) prln(a[a.length - 1]); else prln();}
    static <T> void prln(Collection<T> c) {int n = c.size() - 1; Iterator<T> iter = c.iterator(); for (int i = 0; i < n; pr(iter.next()), pr(' '), ++i); if (n >= 0) prln(iter.next()); else prln();}
    static void h() {prln("hlfd"); flush();}
    static void flush() {__o.flush();}
    static void close() {__o.close();}
}