Jeopardy of Dropped Balls

Description:

Input:

The first line contains three integers $$$n$$$, $$$m$$$, and $$$k$$$ ($$$1 \leq n, m \leq 1000$$$, $$$1 \leq k \leq 10^5$$$) — the size of the grid and the number of balls dropped by Mr. Chanek.

The $$$i$$$-th of the next $$$n$$$ lines contains $$$m$$$ integers $$$a_{i,1},a_{i,2},\ldots,a_{i,m}$$$ ($$$1 \leq a_{i,j} \leq 3$$$). It will satisfy $$$a_{i, 1} \ne 3$$$ and $$$a_{i, m} \ne 1$$$.

The next line contains $$$k$$$ integers $$$c_1, c_2, \ldots, c_k$$$ ($$$1 \leq c_i \leq m$$$) — the balls' column positions dropped by Mr. Chanek sequentially.

Output:

Output $$$k$$$ integers — the $$$i$$$-th integer denoting the column where the $$$i$$$-th ball will end.

Sample Input:

5 5 3
1 2 3 3 3
2 2 2 2 2
2 2 2 2 2
2 2 2 2 2
2 2 2 2 2
1 2 1

Sample Output:

2 2 1 

Sample Input:

1 2 2
1 3
1 2

Sample Output:

1 2 

Note:

In the first example, the first ball will drop as follows. Note that the cell $$$(1, 1)$$$ will change direction to the bottom direction.

The second and third balls will drop as follows.

All balls will be dropped from the first row and on the $$$c_1, c_2, \dots, c_k$$$-th columns respectively. A ball will stop dropping once it leaves the grid.