Problem:
Score : 100 points

Problem Statement

There is a simple directed graph G with N vertices, numbered 1, 2, \ldots, N.

For each i and j (1 \leq i, j \leq N), you are given an integer a_{i, j} that represents whether there is a directed edge from Vertex i to j.
If a_{i, j} = 1, there is a directed edge from Vertex i to j; if a_{i, j} = 0, there is not.

Find the number of different directed paths of length K in G, modulo 10^9 + 7.
We will also count a path that traverses the same edge multiple times.

Constraints

All values in input are integers.

1 \leq N \leq 50

1 \leq K \leq 10^{18}

a_{i, j} is 0 or 1.

a_{i, i} = 0

Input

Input is given from Standard Input in the following format:

N K
a_{1, 1} \ldots a_{1, N}
:
a_{N, 1} \ldots a_{N, N}

Output

Print the number of different directed paths of length K in G, modulo 10^9 + 7.

Sample Input 1

4 2
0 1 0 0
0 0 1 1
0 0 0 1
1 0 0 0

Sample Output 1

6

G is drawn in the figure below:

There are six directed paths of length 2:

1 → 2 → 3

1 → 2 → 4

2 → 3 → 4

2 → 4 → 1

3 → 4 → 1

4 → 1 → 2

Sample Input 2

3 3
0 1 0
1 0 1
0 0 0

Sample Output 2

3

G is drawn in the figure below:

There are three directed paths of length 3:

1 → 2 → 1 → 2

2 → 1 → 2 → 1

2 → 1 → 2 → 3

Sample Input 3

6 2
0 0 0 0 0 0
0 0 1 0 0 0
0 0 0 0 0 0
0 0 0 0 1 0
0 0 0 0 0 1
0 0 0 0 0 0

Sample Output 3

1

G is drawn in the figure below:

There is one directed path of length 2:

4 → 5 → 6

Sample Input 4

1 1
0

Sample Output 4

0

Sample Input 5

10 1000000000000000000
0 0 1 1 0 0 0 1 1 0
0 0 0 0 0 1 1 1 0 0
0 1 0 0 0 1 0 1 0 1
1 1 1 0 1 1 0 1 1 0
0 1 1 1 0 1 0 1 1 1
0 0 0 1 0 0 1 0 1 0
0 0 0 1 1 0 0 1 0 1
1 0 0 0 1 0 1 0 0 0
0 0 0 0 0 1 0 0 0 0
1 0 1 1 1 0 1 1 1 0

Sample Output 5

957538352

Be sure to print the count modulo 10^9 + 7.

Solution in C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include<bits/stdc++.h>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
using namespace std;
using namespace __gnu_pbds;

#define TRACE
#ifdef TRACE
#define tr(...) __f(#__VA_ARGS__, __VA_ARGS__)
template <typename Arg1>
void __f(const char* name, Arg1&& arg1){
    cerr << name << " : " << arg1 << std::endl;
}
template <typename Arg1, typename... Args>
void __f(const char* names, Arg1&& arg1, Args&&... args){
    const char* comma = strchr(names + 1, ',');cerr.write(names, comma - names) << " : " << arg1<<" | ";__f(comma+1, args...);
}
#else
#define tr(...)
#endif

#define ll long long
#define ld long double 
#define endl "\n"
#define PB push_back
#define MP make_pair
#define F first
#define S second
#define SZ(a) (ll)(a.size())
#define all(a) a.begin(), a.end() 
#define fr(i,n) for(i=1;i<=n;i++)
#define rp(i,n) for(i=0;i<n;i++)
#define FIO ios_base::sync_with_stdio(false);cin.tie(0);cout.tie(0);

typedef tree<ll, null_type, less<ll>, rb_tree_tag, tree_order_statistics_node_update> ordered_set;
// order_of_key (val): returns the no. of values less than val
// find_by_order (k): returns the iterator to kth largest element.(0-based)
typedef pair<ll, ll> ii;
typedef vector<ll> vl;
typedef vector<vector<long long> > mat;
const ll N = 200010;
ll mod=1e9+7, n;

ll ffmod(ll x, ll y, ll mod)
{
    ll rt=1;
    while(y)
    {
        if(y&1ll) rt=(rt*x)%mod;
        x=(x*x)%mod, y/=2ll;
    }
    return rt;
}

mat mul(mat v4,mat v5)
{
    mat v6;
    vector<long long> v1;
    int i,j,l,cnt;
    for(i=0;i<n;i++)
    {
        v1.clear();
        for(j=0;j<n;j++)
        {
            cnt=0;
            for(l=0;l<n;l++)
            {
                cnt=(cnt+v4[i][l]*v5[l][j])%mod;
            }
            v1.push_back(cnt);
        }
        v6.push_back(v1);
    }
    return v6;
}

mat power(mat v, ll x)
{
    if(x==1)
        return v;
    mat tp=power(v, x/2);
    if(x%2==1)
        return mul(tp, mul(tp,v));
    return mul(tp,tp);
}

int main()
{
    FIO;
    ll k,x,i,j;
    cin>>n>>k;
    vl v1;
    mat v;
    rp(i,n)
    {
        v1.clear();
        rp(j,n) cin>>x, v1.PB(x);
        v.PB(v1);
    }
    v=power(v,k);
    x=0;
    rp(i,n) rp(j,n) x=(x+v[i][j])%mod;
    cout<<x<<endl;
    return 0;
}