Dark is going to attend Motarack's birthday. Dark decided that the gift he is going to give to Motarack is an array a of n non-negative integers.

Dark created that array 1000 years ago, so some elements in that array disappeared. Dark knows that Motarack hates to see an array that has two adjacent elements with a high absolute difference between them. He doesn't have much time so he wants to choose an integer k (0 ≤ k ≤ 10^{9}) and replaces all missing elements in the array a with k.

Let m be the maximum absolute difference between all adjacent elements (i.e. the maximum value of |a_i - a_{i+1}| for all 1 ≤ i ≤ n - 1) in the array a after Dark replaces all missing elements with k.

Dark should choose an integer k so that m is minimized. Can you help him?

Input

The input consists of multiple test cases. The first line contains a single integer t (1 ≤ t ≤ 10^4) — the number of test cases. The description of the test cases follows.

The first line of each test case contains one integer n (2 ≤ n ≤ 10^{5}) — the size of the array a.

The second line of each test case contains n integers a_1, a_2, …, a_n (-1 ≤ a_i ≤ 10 ^ {9}). If a_i = -1, then the i-th integer is missing. It is guaranteed that at least one integer is missing in every test case.

It is guaranteed, that the sum of n for all test cases does not exceed 4 ⋅ 10 ^ {5}.

Output

Print the answers for each test case in the following format:

You should print two integers, the minimum possible value of m and an integer k (0 ≤ k ≤ 10^{9}) that makes the maximum absolute difference between adjacent elements in the array a equal to m.

Make sure that after replacing all the missing elements with k, the maximum absolute difference between adjacent elements becomes m.

If there is more than one possible k, you can print any of them.

Example

Input


7
5
-1 10 -1 12 -1
5
-1 40 35 -1 35
6
-1 -1 9 -1 3 -1
2
-1 -1
2
0 -1
4
1 -1 3 -1
7
1 -1 7 5 2 -1 5


Output


1 11
5 35
3 6
0 42
0 0
1 2
3 4

Note

In the first test case after replacing all missing elements with 11 the array becomes [11, 10, 11, 12, 11]. The absolute difference between any adjacent elements is 1. It is impossible to choose a value of k, such that the absolute difference between any adjacent element will be ≤ 0. So, the answer is 1.

In the third test case after replacing all missing elements with 6 the array becomes [6, 6, 9, 6, 3, 6].

  * |a_1 - a_2| = |6 - 6| = 0; 
  * |a_2 - a_3| = |6 - 9| = 3; 
  * |a_3 - a_4| = |9 - 6| = 3; 
  * |a_4 - a_5| = |6 - 3| = 3; 
  * |a_5 - a_6| = |3 - 6| = 3. 



So, the maximum difference between any adjacent elements is 3.
import java.io.*;
import java.math.BigInteger;
import java.util.*;
public class Main {
    public static void main(String[] args) throws Exception {
        new Main().run();}
    class BigNumber {

        int len;
        int a[];

        BigNumber(String s) {
            len = s.length();
            a = new int[len];
            for (int i = len - 1, j = 0; i >= 0; --i, ++j) {
                a[j] = s.charAt(i) - '0';
            }
        }

        BigNumber(int[] f) {
            a = f;
            len = a.length;
        }

        BigNumber(int[] f, int nlen) {
            len = nlen;
            a = f;
        }

        int compare(BigNumber s) {
            if (s.len > len) {
                return -1;
            } else if (s.len < len) {
                return 1;
            }
            int b[] = s.a;
            for (int i = len - 1; i >= 0; --i) {
                if (a[i] > b[i]) {
                    return 1;
                } else if (a[i] < b[i]) {
                    return -1;
                }
            }
            return 0;
        }


        BigNumber add(BigNumber s) {
            int b[] = s.a;
            int lenb = s.len;
            int nlen = Math.max(lenb, len) + 1;
            int x[] = new int[nlen];
            int add = 0;
            for (int i = 0; i < nlen; ++i) {
                int sum = add;
                if (i < len) {
                    sum += a[i];
                }
                if (i < lenb) {
                    sum += b[i];
                }
                x[i] = sum % 10;
                add = sum / 10;
            }
            if (x[nlen - 1] == 0) {
                return new BigNumber(x, nlen - 1);
            }
            return new BigNumber(x);
        }

        void mul2(){
            if(a.length==len){
                int x[] = new int[len*2];
                for(int i=0;i<len;++i){
                    x[i]  = a[i];
                }
                len = len*2;
                a = x;
            }
            int add = 0;
            for(int i=0;i<len;i++){
                int v = a[i]*2 + add;
                a[i] = v%10;
                add = v/10;
            }
            if(add>0){
                a[len] = add;
                len++;
            }

        }

        void div2(){
            int p = 0;
            for(int i=len-1;i>=0;--i){
                int v = p *10 + a[i];
                a[i] = v/2;
                p = v%2;
            }
            while(len-1>=0&&a[len-1]==0){
                len--;
            }
            if(len==0){
                len = 1;
            }
        }

        BigNumber substract(BigNumber s) {
            if (this.compare(s) < 0) {
                return s.substract(this);
            }
            int b[] = s.a;
            int lenb = s.len;
            //int x[] = new int[len];
            int minus = 0;
            for (int i = 0; i < len; ++i) {
                int sum = minus + a[i];
                if (i < lenb) {
                    sum -= b[i];
                }
                if (sum < 0) {
                    a[i] = 10 + sum;
                    minus = -1;
                } else {
                    a[i] = sum;
                    minus = 0;
                }
            }
            int l = len;
            while (l - 1 >= 0 && a[l - 1] == 0) {
                --l;
            }
            len = l == 0 ? 1 : l;
            return this;
        }

        public String toString() {
            StringBuilder sb = new StringBuilder();
            for (int i = len - 1; i >= 0; --i) {
                sb.append(a[i]);
            }
            return sb.toString();
        }
    }
//    int[] h,ne,to,wt;
//    int ct = 0;
//    int n;
//    void graph(int n,int m){
//        h = new int[n];
//        Arrays.fill(h,-1);
////        sccno = new int[n];
////        dfn = new int[n];
////        low = new int[n];
////        iscut = new boolean[n];
//        ne = new int[2*m];
//        to = new int[2*m];
//        wt = new int[2*m];
//        ct = 0;
//    }
//    void add(int u,int v,int w){
//        to[ct] = v;
//        ne[ct] = h[u];
//        wt[ct] = w;
//        h[u] = ct++;
//    }
//
//    int color[],dfn[],low[],stack[] = new int[1000000],cnt[];
//    int sccno[];
//    boolean iscut[];
//    int time = 0,top = 0;
//    int scc_cnt = 0;
//
//    // 有向图的强连通分量
//    void tarjan(int u) {
//        low[u] = dfn[u]= ++time;
//        stack[top++] = u;
//        for(int i=h[u];i!=-1;i=ne[i]) {
//            int v = to[i];
//            if(dfn[v]==0) {
//                tarjan(v);
//                low[u]=Math.min(low[u],low[v]);
//            } else if(sccno[v]==0) {
//                // dfn>0 but sccno==0, means it's in current stack
//                low[u]=Math.min(low[u],low[v]);
//            }
//        }
//
//        if(dfn[u]==low[u]) {
//            sccno[u] = ++scc_cnt;
//            while(stack[top-1]!=u) {
//                sccno[stack[top-1]] = scc_cnt;
//                --top;
//            }
//            --top;
//        }
//    }
//
//    //缩点, topology sort
//    int[] h1,to1,ne1;
//    int ct1 = 0;
//    void point(){
//        for(int i=0;i<n;i++) {
//            if(dfn[i]==0) tarjan(i);//有可能图不连通，所以要循环判断。
//        }
//        // 入度
//        int du[] = new int[scc_cnt+1];
//        h1 = new int[scc_cnt+1];
//        Arrays.fill(h1, -1);
//        to1 = new int[scc_cnt*scc_cnt];
//        ne1 = new int[scc_cnt*scc_cnt];
//        // scc_cnt 个点
//
//        for(int i=1;i<=n;i++) {
//            for(int j=h[i]; j!=-1; j=ne[j]) {
//                int y = to[j];
//                if(sccno[i] != sccno[y]) {
//                    // add(sccno[i],sccno[y]);  // 建新图
//                    to1[ct1] = sccno[y];
//                    ne1[ct1] = h[sccno[i]];
//                    h[sccno[i]] = ct1++;
//                    du[sccno[y]]++; //存入度
//                }
//            }
//        }
//
//        int q[] = new int[100000];
//        int end = 0;
//        int st = 0;
//        for(int i=1;i<=scc_cnt;++i){
//            if(du[i]==0){
//                q[end++] = i;
//            }
//        }
//
//        int dp[] = new int[scc_cnt+1];
//        while(st<end){
//            int cur = q[st++];
//            for(int i=h1[cur];i!=-1;i=ne1[i]){
//                int y = to[i];
//                // dp[y] += dp[cur];
//                if(--du[y]==0){
//                    q[end++] = y;
//                }
//            }
//        }
//    }
//
//
//
//
//    int fa[];
//    int faw[];
//
//    int dep = -1;
//    int pt = 0;
//    void go(int rt,int f,int dd){
//
//
//        int p = 0;
//        stk[p] = rt;
//        lk[p] = 0;
//        fk[p] = f;p++;
//        while(p>0) {
//            int cur = stk[p - 1];
//            int fp = fk[p - 1];
//            int ll = lk[p - 1];
//            p--;
//
//
//            if (ll > dep) {
//                dep = ll;
//                pt = cur;
//            }
//            for (int i = h[cur]; i != -1; i = ne[i]) {
//                int v = to[i];
//                if (fp == v) continue;
//
//                stk[p] = v;
//                lk[p] = ll + wt[i];
//                fk[p] = cur;
//                p++;
//            }
//        }
//    }
//    int pt1 = -1;
//    void go1(int rt,int f,int dd){
//
//        int p = 0;
//        stk[p] = rt;
//        lk[p] = 0;
//        fk[p] = f;p++;
//        while(p>0) {
//            int cur = stk[p - 1];
//            int fp = fk[p - 1];
//            int ll = lk[p - 1];
//            p--;
//
//
//            if (ll > dep) {
//                dep = ll;
//                pt1 = cur;
//            }
//
//            fa[cur] = fp;
//            for (int i = h[cur]; i != -1; i = ne[i]) {
//                int v = to[i];
//                if (v == fp) continue;
//                faw[v] = wt[i];
//                stk[p] = v;
//                lk[p] = ll + wt[i];
//                fk[p] = cur;
//                p++;
//            }
//        }
//    }
//
//    int r = 0;
//    int stk[] = new int[301];
//    int fk[] = new int[301];
//    int lk[] = new int[301];
//    void ddfs(int rt,int t1,int t2,int t3,int l){
//
//
//        int p = 0;
//        stk[p] = rt;
//        lk[p] = 0;
//        fk[p] = t3;p++;
//        while(p>0){
//            int cur = stk[p-1];
//            int fp = fk[p-1];
//            int ll = lk[p-1];
//            p--;
//            r = Math.max(r,ll);
//            for(int i=h[cur];i!=-1;i=ne[i]){
//                int v = to[i];
//                if(v==t1||v==t2||v==fp) continue;
//                stk[p] = v;
//                lk[p] = ll+wt[i];
//                fk[p] = cur;p++;
//            }
//        }
//
//
//
//    }


    static long mul(long a, long b, long p)
    {
        long res=0,base=a;
        while(b>0)
        {
            if((b&1L)>0)
                res=(res+base)%p;
            base=(base+base)%p;
            b>>=1;
        }
        return res;
    }

    static long mod_pow(long k,long n,long p){
        long res = 1L;
        long temp = k;
        while(n!=0L){
            if((n&1L)==1L){
                res = (res*temp)%p;
            }
            temp = (temp*temp)%p;
            n = n>>1L;
        }
        return res%p;
    }

    void sort(long a[]) {
        Random rd = new Random();
        for (int i = 1; i < a.length; ++i) {
            int p = rd.nextInt(i); long x = a[p]; a[p] = a[i]; a[i] = x;
        }
        Arrays.sort(a);
    }

    void dfs(int from,int k){

    }
    int ct = 0;
    void add(int u,int v,int w){
        to[ct] = v;
        wt[ct] = w;
        ne[ct] = h[u];
        h[u] = ct++;
    }
    int r =0;
    void dfs1(int c,int ff){
        clr[c][aa[c]]++;
        for(int j=h[c];j!=-1;j=ne[j]){
            if(to[j]==ff) continue;
            dfs1(to[j],c);
            clr[c][1] += clr[to[j]][1];
            clr[c][2] += clr[to[j]][2];
            if(clr[to[j]][1]==s1&&clr[to[j]][2]==0||clr[to[j]][2]==s2&&clr[to[j]][1]==0){
                r++;
            }
        }
    }

    int[] h,ne,to,fa1,wt;
    int clr[][];
    int aa[];
    int s1 = 0;
    int s2 = 0;

    boolean f(int n){
        int c = 0;
        while(n>0){
            c += n%10;
            n /=10;
        }
        return (c&3)==0;
    }

    int[][] next(String s){
        int len = s.length();
        int ne[][] = new int[len+1][26];
        Arrays.fill(ne[len], -1);
        for(int i=len-1;i>=0;--i){
            ne[i] = ne[i+1].clone();
            ne[i][s.charAt(i)-'a'] = i+1;
        }
        return ne;
    }


    void sort(int a[]){
        Random rd =new Random();
        for(int i=1;i<a.length;++i){
            int id = rd.nextInt(i);
            int t = a[id];
            a[id] = a[i];
            a[i] = t;
        }
    }

    boolean same(String s){
        return s.charAt(0)==s.charAt(1);
    }
    boolean next_perm(int[] a){
        int len = a.length;
        for(int i=len-2,j = 0;i>=0;--i){
            if(a[i]<a[i+1]){
                j = len-1;
                for(;a[j]<=a[i];--j);
                int p = a[j];
                a[j] = a[i];
                a[i] = p;
                j = i+1;
                for(int ed = len-1;j<ed;--ed) {
                    p = a[ed];
                    a[ed] = a[j];
                    a[j++] = p;
                }
                return true;
            }
        }
        return false;
    }

    void print1(int c[],int n){
        StringBuilder sb = new StringBuilder();
        for(int i=0;i<n;++i){
            for(int j=0;j<3;++j) {
                sb.appendCodePoint(c[j]+'a');
            }
        }
        println(sb);
    }
    void print2(int c[],int n){
        StringBuilder sb = new StringBuilder();
        for(int j=0;j<3;++j) {
            for (int i = 0; i < n; ++i) {
                sb.appendCodePoint(c[j]+'a');
            }
        }
        println(sb);
    }


    int[] dp(int a[][],int ck){
        left = new int[n+1][m+1];
        right = new int[n+1][m+1];
        up = new int[n+1][m+1];

        for(int i=1;i<=n;++i){
            for(int j=1;j<=m;++j){
                if(a[i][j]==ck){
                    left[i][j] = right[i][j] = j;
                    up[i][j] = 1;
                }
            }
        }

        for(int i=1;i<=n;++i){
            for(int j=2;j<=m;++j){
                if(a[i][j]==ck&&ck==a[i][j-1]) {
                    left[i][j] = left[i][j-1];
                }
            }
        }

        for(int i=1;i<=n;++i){
            for(int j=m-1;j>=1;--j){
                if(a[i][j]==ck&&ck==a[i][j+1]) {
                    right[i][j] = right[i][j+1];
                }
            }
        }

//        for(int i=1;i<=n;++i) {
//            for (int j = 1; j <= m; ++j) {
//                print(right[i][j]);
//            }
//            println();
//        }


        int all1 = 0;
        int all2 = 0;

        for(int i=1;i<=n;++i) {
            for (int j = 1; j <= m; ++j) {
                if(i>=2&&a[i][j]==ck&&ck==a[i-1][j]) {
                    up[i][j] = up[i-1][j] + 1;
                    left[i][j] = Math.max(left[i-1][j] ,left[i][j]);
                    right[i][j] = Math.min(right[i-1][j] ,right[i][j]);
                }
                if(a[i][j]==ck) {
                    int s2 = up[i][j] * (right[i][j] - left[i][j] + 1);
                    int u = Math.min(up[i][j], right[i][j] - left[i][j] + 1);
                    if (all2 < s2) {
                        all2 = s2;
                    }
                    if (all1 < u*u ) {
                        all1 = u*u;
                    }
                }
            }
        }

        return new int[]{all1,all2};




    }
    int up[][],right[][],left[][];
    public static String roundS(double result, int scale){
        String fmt = String.format("%%.%df", scale);
        return String.format(fmt, result);
    }

    int n;int m;


    static int MAXN = 100000;
    static int tree[] = new int[MAXN];

    void init1(int f[]){
        for(int i=1;i<tree.length;++i){
            tree[i] += f[i];
            if(i+(i&-i)<tree.length){
                tree[i+(i&-i)] += tree[i];
            }
        }
    }


    void update(int i,int v){
        while(i<tree.length){
            tree[i] += v;
            i += i&(-i);
        }
    }

    int sum(int i){
        int v = 0;
        while(i>0){
            v = v + tree[i];
            i -= i&(-i);
        }
        return v;
    }

    int equal_bigger(int v){
        int s=0,p=0;
        for(int i=Integer.numberOfTrailingZeros(Integer.highestOneBit(tree.length));i>=0;--i) {
            if(p+(1<<i)< tree.length && s + tree[p+(1<<i)] <v){
                v -= tree[p+(1<<i)];
                p |= 1<<i;
            }
        }
        return p+1;
    }

    int ct1 = 0;
    int mx = 0;
    void dfs(String s,int cur,int p){
        int fk = s.charAt(p)-'0';
        if(ct1==m){
            return;
        }
        if(p>=1){
            ct1++;
            mx = Math.max(cur,mx);
            if(ct1==m){
                return;
            }
        }
        for(int i=p>0?0:1;i<=fk;++i){
            dfs(s,cur*10+i,p+1);
        }
    }


    long yj(int i,int j){

        long r = yjk(i,j);
        if(i==j){
            r += 1;
        }
        if(j>=i) {
            r += z(i, j - 1);
        }
        return r;
    }

    long yjk(int i,int j){
        if(j>=i+1){
            return pd(i+1,j-1);
        }else{
            return 0;
        }
    }

    long z(int i,int j){
        long r = hs(i,j);
        if(j>=i+1){
            r += z(i+1,j-1)+ hs(i,j-2);
        }
        return r;
    }

    long hs(int i,int j){
        long r = 0;
        if(j>=i+1){
            r += hs(i+1,j-1) + yjk(i+2,j);
        }
        if(j>=i+3){
            r += yjk(i+4,j);
        }
        return r;
    }



    long pd(int i,int j){
        if(i==j+1) return 1;
        if(i>j) return 0;
        long r = 0;
        for(int k=i;k<=j;++k){
            long mb = pd(i,k-1)*yj(k,j);
            r += mb;
        }
        return r;
    }




    void solve() {

       for(int i = ni();i>0;--i){
           int n = ni();
           long a[] = nal(n);
           long mx = -2;
           long mi = 2000000000L;
           for(int j=1;j<n;++j){
               if(a[j]==-1&&a[j-1]!=-1){
                   mx = Math.max(mx,a[j-1]);
                   mi = Math.min(mi,a[j-1]);
               }else if(a[j-1]==-1&&a[j]!=-1){
                   mx = Math.max(mx,a[j]);
                   mi = Math.min(mi,a[j]);
               }
           }
           long d =  (mx+mi)/2L;
           for(int j=0;j<n;++j) {
               if(a[j]==-1) {
                   a[j] = d;
               }
           }
           long r = 0L;
           for(int j=1;j<n;++j){
               r = Math.max(r,Math.abs(a[j]-a[j-1]));
           }
           println(r+" "+d);
       }















//        for(int i=500;i>=100;--i){
//            if(good(i)!=bad(i)){
//                println(good(i));
//                println(bad(i));
//                println(i);
//                println("-----------------------");
//            }
//        }





































        //N , M , K , a , b , c , d . 其中N , M是矩阵的行列数；K 是上锁的房间数目，(a, b)是起始位置，(c, d)是出口位置


//        int n = ni();
//        int m = ni();
//        int k = ni();
//        int a = ni();
//        int b = ni();
//        int c = ni();
//        int d = ni();
//
//
//        char cc[][] = nm(n,m);
//        char keys[][] = new char[n][m];
//
//        char ky = 'a';
//        for(int i=0;i<k;++i){
//            int x = ni();
//            int y = ni();
//            keys[x][y] = ky;
//            ky++;
//        }
//        int f1[] = {a,b,0};
//
//        int dd[][] = {{0,1},{0,-1},{1,0},{-1,0}};
//
//        Queue<int[]> q =  new LinkedList<>();
//        q.offer(f1);
//        int ts = 1;
//
//        boolean vis[][][] = new boolean[n][m][33];
//
//        while(q.size()>0){
//            int sz = q.size();
//            while(sz-->0) {
//                int cur[] = q.poll();
//                vis[cur[0]][cur[1]][cur[2]] = true;
//
//                int x = cur[0];
//                int y = cur[1];
//
//                for (int u[] : dd) {
//                       int lx = x +  u[0];
//                       int ly = y +  u[1];
//                       if (lx >= 0 && ly >= 0 && lx < n && ly < m && (cc[lx][ly] != '#')&&!vis[lx][ly][cur[2]]){
//                            char ck =cc[lx][ly];
//                            if(ck=='.'){
//                                if(lx==c&&ly==d){
//                                    println(ts); return;
//                                }
//                                if(keys[lx][ly]>='a'&&keys[lx][ly]<='z') {
//                                    int cao = cur[2] | (1 << (keys[lx][ly] - 'a'));
//                                    q.offer(new int[]{lx, ly, cao});
//                                    vis[lx][ly][cao] = true;
//                                }else {
//
//                                    q.offer(new int[]{lx, ly, cur[2]});
//                                }
//
//                            }else if(ck>='A'&&ck<='Z'){
//                                int g = 1<<(ck-'A');
//                                if((g&cur[2])>0){
//                                    if(lx==c&&ly==d){
//                                        println(ts); return;
//                                    }
//                                    if(keys[lx][ly]>='a'&&keys[lx][ly]<='z') {
//
//                                        int cao = cur[2] | (1 << (keys[lx][ly] - 'a'));
//                                        q.offer(new int[]{lx, ly, cao});
//                                        vis[lx][ly][cao] = true;;
//                                    }else {
//
//                                        q.offer(new int[]{lx, ly, cur[2]});
//                                    }
//                                }
//                            }
//                       }
//                }
//
//                }
//            ts++;
//        }
//        println(-1);




//        int n = ni();
//
//        HashSet<String> st = new HashSet<>();
//        HashMap<String,Integer> mp = new HashMap<>();
//
//
//        for(int i=0;i<n;++i){
//            String s  = ns();
//            int id= 1;
//            if(mp.containsKey(s)){
//                int u = mp.get(s);
//                id = u;
//
//            }
//
//            if(st.contains(s)) {
//
//                while (true) {
//                    String ts = s + id;
//                    if (!st.contains(ts)) {
//                        s = ts;
//                        break;
//                    }
//                    id++;
//                }
//                mp.put(s,id+1);
//            }else{
//                mp.put(s,1);
//            }
//            println(s);
//            st.add(s);
//
//        }

//        int t = ni();
//
//        for(int i=0;i<t;++i){
//            int n = ni();
//            long w[] = nal(n);
//
//            Map<Long,Long> mp = new HashMap<>();
//            PriorityQueue<long[]> q =new PriorityQueue<>((xx,xy)->{return Long.compare(xx[0],xy[0]);});
//
//            for(int j=0;j<n;++j){
//                q.offer(new long[]{w[j],0});
//                mp.put(w[j],mp.getOrDefault(w[j],0L)+1L);
//            }
//
//            while(q.size()>=2){
//                long f[] = q.poll();
//                long y1 = f[1];
//                if(y1==0){
//                    y1 = mp.get(f[0]);
//                    if(y1==1){
//                        mp.remove(f[0]);
//                    }else{
//                        mp.put(f[0],y1-1);
//                    }
//                }
//                long g[] = q.poll();
//                long y2 = g[1];
//                if(y2==0){
//                    y2 = mp.get(g[0]);
//                    if(y2==1){
//                        mp.remove(g[0]);
//                    }else{
//                        mp.put(g[0],y2-1);
//                    }
//                }
//                q.offer(new long[]{f[0]+g[0],2L*y1*y2});
//
//            }
//            long r[] = q.poll();
//            println(r[1]);
//
//
//
//
//        }
        //  int o= 9*8*7*6;
        //  println(o);


//       int t = ni();
//       for(int i=0;i<t;++i){
//           long a = nl();
//           int k  = ni();
//           if(k==1){
//               println(a);
//               continue;
//           }
//
//           int f = (int)(a%10L);
//           int s = 1;
//           int j = 0;
//           for(;j<30;j+=2){
//               int u = f-j;
//               if(u<0){
//                   u = 10+u;
//               }
//               s = u*s;
//               s = s%10;
//               if(s==k){
//                   break;
//               }
//           }
//
//           if(s==k) {
//               println(a - j - 2);
//           }else{
//               println(-1);
//           }
//
//
//
//
//       }

//     int m = ni();
//     h = new int[n];
//     to = new int[2*(n-1)];
//     ne = new int[2*(n-1)];
//     wt = new int[2*(n-1)];
//
//     for(int i=0;i<n-1;++i){
//         int u = ni()-1;
//         int v = ni()-1;
//
//     }
        //  long a[] = nal(n);


//        int n = ni();
//        int k = ni();
//        t1 = new long[200002];
//
//        int p[][] = new int[n][3];
//
//        for(int i=0;i<n;++i){
//            p[i][0]  = ni();
//            p[i][1]  = ni();
//            p[i][2]  = i+1;
//        }
//        Arrays.sort(p, new Comparator<int[]>() {
//            @Override
//            public int compare(int[] x, int[] y) {
//                if(x[1]!=y[1]){
//                    return Integer.compare(x[1],y[1]);
//                }
//                return Integer.compare(y[0], x[0]);
//            }
//        });
//
//        for(int i=0;i<n;++i){
//            int ck = p[i][0];
//
//        }

















    }









//    int []h,to,ne,wt;


    //    long t2[];
    void update(long[] t,int i,long v){
        for(;i<t.length;i+=(i&-i)){
            t[i] += v;
        }
    }
    long get(long[] t,int i){
        long s = 0;
        for(;i>0;i-=(i&-i)){
            s += t[i];
        }
        return s;
    }

    int equal_bigger(long t[],long v){
        int s=0,p=0;
        for(int i=Integer.numberOfTrailingZeros(Integer.highestOneBit(t.length));i>=0;--i) {
            if(p+(1<<i)< t.length && s + t[p+(1<<i)] < v){
                v -= t[p+(1<<i)];
                p |= 1<<i;
            }
        }
        return p+1;
    }









    static class S{
        int l = 0;
        int r = 0 ;
        long le = 0;
        long ri = 0;
        long tot = 0;
        long all = 0;

        public S(int l,int r) {
            this.l = l;
            this.r = r;
        }
    }

    static S a[];
    static int[] o;

    static void init(int[] f){
        o = f;
        int len = o.length;
        a = new S[len*4];
        build(1,0,len-1);
    }

    static void build(int num,int l,int r){
        S cur = new S(l,r);
        if(l==r){
            a[num] = cur;
            return;
        }else{
            int m = (l+r)>>1;
            int le = num<<1;
            int ri = le|1;
            build(le, l,m);
            build(ri, m+1,r);
            a[num] = cur;
            pushup(num, le, ri);
        }
    }



    //    static int query(int num,int l,int r){
//
//        if(a[num].l>=l&&a[num].r<=r){
//            return a[num].tot;
//        }else{
//            int m = (a[num].l+a[num].r)>>1;
//            int le = num<<1;
//            int ri = le|1;
//            pushdown(num, le, ri);
//            int ma = 1;
//            int mi = 100000001;
//            if(l<=m) {
//                int r1 = query(le, l, r);
//                ma = ma*r1;
//
//            }
//            if(r>m){
//                int r2 = query(ri, l, r);
//                ma = ma*r2;
//            }
//            return ma;
//        }
//    }
    static long dd = 10007;
    static void update(int num,int l,long v){
        if(a[num].l==a[num].r){
            a[num].le = v%dd;
            a[num].ri = v%dd;
            a[num].all = v%dd;
            a[num].tot = v%dd;
        }else{
            int m = (a[num].l+a[num].r)>>1;
            int le = num<<1;
            int ri = le|1;
            pushdown(num, le, ri);
            if(l<=m){
                update(le,l,v);
            }
            if(l>m){
                update(ri,l,v);
            }
            pushup(num,le,ri);
        }
    }

    static void pushup(int num,int le,int ri){
        a[num].all = (a[le].all*a[ri].all)%dd;
        a[num].le = (a[le].le + a[le].all*a[ri].le)%dd;
        a[num].ri = (a[ri].ri + a[ri].all*a[le].ri)%dd;
        a[num].tot = (a[le].tot + a[ri].tot + a[le].ri*a[ri].le)%dd;


        //a[num].res[1] = Math.min(a[le].res[1],a[ri].res[1]);
    }
    static void pushdown(int num,int le,int ri){

    }

    long gcd(long a,long b){ return b==0?a: gcd(b,a%b);}
    int gcd(int a,int b){ return b==0?a: gcd(b,a%b);}
    InputStream is;PrintWriter out;
    void run() throws Exception {is = System.in;
        // is  = new FileInputStream("C:\\Users\\Luqi\\Downloads\\P1273_2.in");
        out = new PrintWriter(System.out);solve();out.flush();}
    private byte[] inbuf = new byte[2];
    public int lenbuf = 0, ptrbuf = 0;
    private int readByte() {
        if (lenbuf == -1) throw new InputMismatchException();
        if (ptrbuf >= lenbuf) {
            ptrbuf = 0;
            try {lenbuf = is.read(inbuf);} catch (IOException e) {throw new InputMismatchException();}
            if (lenbuf <= 0) return -1;
        }
        return inbuf[ptrbuf++];}
    private boolean isSpaceChar(int c) {return !(c >= 33 && c <= 126);}
    private int skip() {int b;while((b = readByte()) != -1 && isSpaceChar(b));return b;}
    private double nd() {return Double.parseDouble(ns());}
    private char nc() {return (char) skip();}
    private char ncc() {int b;while((b = readByte()) != -1 && !(b >= 32 && b <= 126));return (char)b;}
    private String ns() {int b = skip();StringBuilder sb = new StringBuilder();
        while (!(isSpaceChar(b))) { // when nextLine, (isSpaceChar(b) && b != ' ')
            sb.appendCodePoint(b);b = readByte(); }
        return sb.toString();}
    private char[] ns(int n) {char[] buf = new char[n];int b = skip(), p = 0;
        while (p < n && !(isSpaceChar(b))) { buf[p++] = (char) b;b = readByte(); }
        return n == p ? buf : Arrays.copyOf(buf, p);}
    private String nline() {int b = skip();StringBuilder sb = new StringBuilder();
        while (!isSpaceChar(b) || b == ' ') { sb.appendCodePoint(b);b = readByte(); }
        return sb.toString();}
    private char[][] nm(int n, int m) {char[][] a = new char[n][];for (int i = 0; i < n; i++) a[i] = ns(m);return a;}
    private int[] na(int n) {int[] a = new int[n];for (int i = 0; i < n; i++) a[i] = ni();return a;}
    private long[] nal(int n) { long[] a = new long[n];for (int i = 0; i < n; i++) a[i] = nl();return a;}
    private int ni() { int num = 0, b; boolean minus = false;
        while ((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-')){};
        if (b == '-') { minus = true; b = readByte(); }
        while (true) {
            if (b >= '0' && b <= '9') num = (num << 3) + (num << 1) + (b - '0');
            else return minus ? -num : num;
            b = readByte();}}
    private long nl() { long num = 0; int b;
        boolean minus = false;
        while ((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-')){};
        if (b == '-') { minus = true; b = readByte(); }
        while (true) {
            if (b >= '0' && b <= '9')  num = num * 10 + (b - '0');
            else return minus ? -num : num;
            b = readByte();}}
    void print(Object obj){out.print(obj);}
    void println(Object obj){out.println(obj);}
    void println(){out.println();}
}