Split an $a \times b$ numerical matrix as follows: split the original matrix along a straight line into two matrices, then continue to split the resulting matrices in the same way (it is also possible to split only one of them). After performing this split $(n-1)$ times, the original matrix is divided into $n$ matrices. (Each split can only be made along the gaps between the numbers.)

Each position in the original matrix has a value, and the total score of a matrix is the sum of the values at all positions it contains. Now, we need to divide the matrix into $n$ matrices according to the above rules such that the mean square error of the total scores of the $n$ matrices is minimized.

Write a program to find the minimum mean square error for the given matrix and $n$.

Input

The first line contains three integers $a, b, n$ ($1 < a, b \le 10, 1 < n \le 10$).

The next $a$ lines each contain $b$ non-negative integers less than $100$, representing the values at the corresponding positions in the matrix. Adjacent numbers in each line are separated by a space.

Output

A single number, which is the minimum mean square error (rounded to 2 decimal places).

Examples

Input 1

5 4 4
2 3 4 6
5 7 5 1
10 4 0 5
2 0 2 3
4 1 1 1

Output 1

0.50