Problem Statement

Problem F and F2 are the same problem, but with different constraints and time limits.

We have a board divided into $N$ horizontal rows and $N$ vertical columns of square cells. The cell at the $i$-th row from the top and the $j$-th column from the left is called Cell $(i,j)$. Each cell is either empty or occupied by an obstacle. Also, each empty cell has a digit written on it. If $A_{i,j}=$ 1, 2, ..., or 9, Cell $(i,j)$ is empty and the digit $A_{i,j}$ is written on it. If $A_{i,j}=$ #, Cell $(i,j)$ is occupied by an obstacle.

Cell $Y$ is reachable from cell $X$ when the following conditions are all met:

• Cells $X$ and $Y$ are different.
• Cells $X$ and $Y$ are both empty.
• One can reach from Cell $X$ to Cell $Y$ by repeatedly moving right or down to an adjacent empty cell.

Consider all pairs of cells $(X,Y)$ such that cell $Y$ is reachable from cell $X$. Find the sum of the products of the digits written on cell $X$ and cell $Y$ for all of those pairs.

Constraints

• $1 \leq N \leq 1500$
• $A_{i,j}$ is one of the following characters: 1, 2, ... 9 and #.

Input

Input is given from Standard Input in the following format:

N
A_{1,1}A_{1,2}...A_{1,N}
A_{2,1}A_{2,2}...A_{2,N}
:
A_{N,1}A_{N,2}...A_{N,N}

Output

Print the sum of the products of the digits written on cell $X$ and cell $Y$ for all pairs $(X,Y)$ such that cell $Y$ is reachable from cell $X$.