Out of Boundary - Transition

From $(r, c)$ with $k$ moves, you can go up, down, left, or right. The formula: $dp[r][c][k] = dp[r-1][c][k-1] + dp[r+1][c][k-1] + dp[r][c-1][k-1] + dp[r][c+1][k-1]$ Base case: if $(r, c)$ is outside the grid, return $1$. That path successfully left.

If $k = 0$ and you're still inside, return $0$. No moves left, didn't escape. Don't forget: take results modulo $10^9 + 7$ since counts can get huge. Each cell receives probability from its four neighbors. The grid acts like a probability distribution over positions.