Bài này là bài V11PLAN với giới hạn N lớn hơn.

Note: this is V11PLAN with higher limit for N.

In 2011, Vietnam sets out a national development plan. The plan will consist of two phases: the first half of the year and the second half of the year. At each phase, a number of paths between some pairs of cities will be built.

You are given a graph describle the result of the plan, in which each edge (i,j) represents the plan to make city i and city j connected (not necessarily directly). You need to count the number of different plans that can produce that graph. Two plans are  different if there is a road being built in a phase of the plan but not built in the corresponding phase of the other plan.

For example, if we build in the first phase road (1,2) , and then in the second phase we build (2,3) , the resulting graph will have three edges: (1,2), (2,3), (1,3). Building (1,2) in the first phase and (1,2), (1,3) in the later phase produce the same result. Note, we can only build a road between a pair of cities in a phase, but at the later phase we can rebuild that route again due to subsidence rate in Vietnam is pretty fast.

Input

The first line is N (1 <= N <= 80).

N lines follow, each has N integers. City i and j should be connected at the end of the year if the j-th number at line i is 1, else it's 0. The input will make sure that is city i and j are connected, j and k are connected, then i and k are connected.

Output

A single integer which is the number of different plans modulo 1000000007.

Example

Input:
2
0 1
1 0
Output:
3

Explain: we can build the road between two cities only in the first haft of the year, second half of the year, or both.

Input:
3
0 1 1
1 0 1
1 1 0
Output:
54

Input:
16
0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0
Output:
604876153