Score : $1500$ points

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 $X$.

2
11
11

5

There are five pairs of cells $(X,Y)$ such that cell $Y$ is reachable from cell $X$, as follows:

• $X=(1,1)$, $Y=(1,2)$
• $X=(1,1)$, $Y=(2,1)$
• $X=(1,1)$, $Y=(2,2)$
• $X=(1,2)$, $Y=(2,2)$
• $X=(2,1)$, $Y=(2,2)$

The product of the digits written on cell $X$ and cell $Y$ is $1$ for all of those pairs, so the answer is $5$.

4
1111
11#1
1#11
1111

47

10
76##63##3#
8445669721
75#9542133
3#285##445
749632##89
2458##9515
5952578#77
1#3#44196#
4355#99#1#
#298#63587

36065

10
4177143673
7#########
5#1716155#
6#4#####5#
2#3#597#6#
6#9#8#3#5#
5#2#899#9#
1#6#####6#
6#5359657#
5#########

6525