Package for this problem was not updated by the problem writer or Codeforces administration after we’ve upgraded the judging servers. To adjust the time limit constraint, solution execution time will be multiplied by 2. For example, if your solution works for 400 ms on judging servers, then value 800 ms will be displayed and used to determine the verdict.

Virtual contest is a way to take part in past contest, as close as possible to participation on time. It is supported only ACM-ICPC mode for virtual contests.
If you've seen these problems, a virtual contest is not for you - solve these problems in the archive.
If you just want to solve some problem from a contest, a virtual contest is not for you - solve this problem in the archive.
Never use someone else's code, read the tutorials or communicate with other person during a virtual contest.

No tag edit access

B. Distinct Paths

time limit per test

2 secondsmemory limit per test

256 megabytesinput

standard inputoutput

standard outputYou have a rectangular *n* × *m*-cell board. Some cells are already painted some of *k* colors. You need to paint each uncolored cell one of the *k* colors so that any path from the upper left square to the lower right one doesn't contain any two cells of the same color. The path can go only along side-adjacent cells and can only go down or right.

Print the number of possible paintings modulo 1000000007 (10^{9} + 7).

Input

The first line contains three integers *n*, *m*, *k* (1 ≤ *n*, *m* ≤ 1000, 1 ≤ *k* ≤ 10). The next *n* lines contain *m* integers each — the board. The first of them contains *m* uppermost cells of the board from the left to the right and the second one contains *m* cells from the second uppermost row and so on. If a number in a line equals 0, then the corresponding cell isn't painted. Otherwise, this number represents the initial color of the board cell — an integer from 1 to *k*.

Consider all colors numbered from 1 to *k* in some manner.

Output

Print the number of possible paintings modulo 1000000007 (10^{9} + 7).

Examples

Input

2 2 4

0 0

0 0

Output

48

Input

2 2 4

1 2

2 1

Output

0

Input

5 6 10

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

Output

3628800

Input

2 6 10

1 2 3 4 5 6

0 0 0 0 0 0

Output

4096

Codeforces (c) Copyright 2010-2019 Mike Mirzayanov

The only programming contests Web 2.0 platform

Server time: Mar/22/2019 16:12:47 (d1).

Desktop version, switch to mobile version.

Supported by

User lists

Name |
---|