## USACO 2020 December Contest, Platinum

## Problem 2. Spaceship

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The aliens will release Bessie if she can complete a strange task. First, they will choose two rooms, $s$ and $t$ $(1 \le s, t \le N)$, and two numbers, $b_s$ and $b_t$ $(1 \le b_s, b_t \le K)$. They will start Bessie in room $s$ and immediately have her press button $b_s$. Bessie will then proceed to navigate the ship while pressing buttons. There are a few rules for what Bessie can do:

- In each room, after pressing exactly one button, she must choose to either exit through a door to another (possibly the same) room or stop.
- Once Bessie presses a button, it is invalid for her to press the same button again unless, in the time between uses, she has pressed a button with a higher number. In other words, pressing button number $x$ will make it unavailable for use, while all buttons with numbers $<x$ will be reset and again available for use.
- If Bessie presses an invalid button, she automatically fails and the aliens will keep her.

Bessie is released only if she stops in room $t$, the last button she pressed was $b_t$, and no invalid buttons were ever pressed.

Bessie is worried that she may not be able to complete the task. For $Q$ $(1\le Q\le 60)$ queries, each consisting of what Bessie considers a likely choice of $s, t, b_s$, and $b_t$, Bessie wants to know the number of sequences of rooms and button presses that would lead to her release. Report your answers modulo $10^9 + 7$ as they may be very large.

#### INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains $N,K,Q$.The next $N$ lines each contain $N$ bits (each 0 or 1). The $j$-th entry of the $i$-th line is 1 if there exists a door from room $i$ to room $j$, and 0 if no such door exists.

This is followed by $Q$ lines, each containing four integers $b_s$, $s$, $b_t$, $t$, denoting the starting button, starting room, final button, and final room respectively.

#### OUTPUT FORMAT (print output to the terminal / stdout):

The number of sequences for each of the $Q$ queries modulo $10^9+7$ on separate lines.#### SAMPLE INPUT:

6 3 8 010000 001000 000100 000010 000000 000001 1 1 1 1 3 3 1 1 1 1 3 3 1 1 1 5 2 1 1 5 1 1 2 5 3 1 3 5 2 6 2 6

#### SAMPLE OUTPUT:

1 0 1 3 2 2 0 5

The doors connect rooms $1\to 2$, $2 \to 3$, $3\to 4$, $4\to 5$, and $6\to 6$.

For the first query, Bessie must stop immediately after pressing the first button.

For the second query, the answer is clearly zero because there is no way to get to room 1 from room 3.

For the third query, Bessie's only option is to move from room 1 to room 2 to room 3 while pressing buttons 1, 2, and 3.

For the fourth query, Bessie's pattern of movement is fixed, and she has three possible sequences of button presses:

- $(1,2,3,2,1)$
- $(1,2,1,3,1)$
- $(1,3,1,2,1)$

For the last query, Bessie has five possible sequences of button presses:

- $(2)$
- $(2,3,2)$
- $(2,3,1,2)$
- $(2,1,3,2)$
- $(2,1,3,1,2)$

#### SAMPLE INPUT:

6 4 6 001100 001110 101101 010111 110111 000111 3 2 4 3 3 1 4 4 3 4 4 1 3 3 4 3 3 6 4 3 3 1 4 2

#### SAMPLE OUTPUT:

26 49 29 27 18 22

This test case satisfies the constraints for all subtasks aside from the first.

#### SAMPLE INPUT:

6 10 5 110101 011001 001111 101111 111010 000001 2 5 2 5 6 1 5 2 3 4 8 3 9 3 3 5 5 1 3 4

#### SAMPLE OUTPUT:

713313311 716721076 782223918 335511486 539247783

Make sure to output the answers modulo $10^9+7$.

#### SCORING:

- In test cases 4-7, $K\le 5$ and $(b_s,s)$ is the same for all queries.
- In test cases 8-11, $b_s=K-1$ and $b_t=K$ for each query.
- In test cases 12-15, $N,K,Q\le 20$.
- In test cases 16-23, there are no additional constraints.

Problem credits: Benjamin Qi