1. **Problem Statement:** We have a grid $a$ with $n$ rows and $m$ columns, each cell $(i,j)$ containing a non-negative integer $a_{i,j}$. We define a path $p$ of length $k$ as a sequence of distinct adjacent cells. The number of turns $t(p)$ in path $p$ counts indices where the path changes direction diagonally. 2. **Conditions for a valid path $p=[(x_0,y_0),\ldots,(x_k,y_k)]$:** - $k \geq 1$ - $t(p) \leq 2$ - $a_{x_0,y_0} = a_{x_k,y_k} > 0$ - For all intermediate cells $1 \leq i < k$, $a_{x_i,y_i} = 0$ 3. **Goal:** Find the number of unordered pairs of cells $(u_1,v_1)$ and $(u_2,v_2)$ that are connectable by such a valid path. 4. **Key observations:** - The path starts and ends on cells with the same positive integer. - Intermediate cells must be zero. - The path can have at most 2 turns. 5. **Approach:** - Group cells by their positive integer values. - For each group, find pairs of cells that can be connected by a path with at most 2 turns through zero cells. - Since the path can have at most 2 turns, it can be: - A straight line (0 turns) - One bend (1 turn) - Two bends (2 turns) 6. **Checking connectivity:** - For each pair of cells with the same positive integer, check if a path with $\leq 2$ turns exists passing only through zeros (except endpoints). - This involves checking lines and L-shaped or U-shaped paths. 7. **Counting pairs:** - Count all such pairs for all positive integers. **Final answer:** The number of connectable pairs is the count of all unordered pairs of cells with the same positive integer value connected by a valid path as defined. (Note: The problem is combinatorial and algorithmic; the exact numeric answer depends on the input grid.)