Pipeline Plans

There are twelve types of tiles in Fig. 1. You were asked to fill a table with R × C cells with these tiles. R is the number of rows and C is the number of columns.

How many arrangements in the table meet the following constraints?

• Each cell has one tile.

• the center of the upper left cell (1,1) and the center of the lower right cell (C, R) are connected by some roads.

Fig. 1: the types of tiles

The first line contains two integers R and C (2 ≤ R × C ≤ 15). You can safely assume at least one of R and C is greater than 1.

The second line contains twelve integers, t1, t2, ..., t12 (0 ≤ t1 + .... + t12 ≤ 15). ti represents the number of the i-th tiles you have.

Output the number of arrangments in a line.