Matrix Walk

Given an $n \times m$ matrix $a$ consisting of only positive integers. You have to pick a $1 \leq j \leq m$, and start from position $(1, j)$. You maintain a sum, initially equal to $a_{1,j}$. You keep adding numbers you land upon. You can make a total of $p$ moves of the following type. In one move you can move either down or up (while staying within the grid) and add the number you land upon to