Mr. Chanek's city can be represented as a plane. He wants to build a housing complex in the city.
There are some telephone poles on the plane, which is represented by a grid \(a\) of size \((n + 1) \times (m + 1)\). There is a telephone pole at \((x, y)\) if \(a_{x, y} = 1\).
For each point \((x, y)\), define \(S(x, y)\) as the square of the Euclidean distance between the nearest pole and \((x, y)\). Formally, the square of the Euclidean distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \((x_2 - x_1)^2 + (y_2 - y_1)^2\).
To optimize the building plan, the project supervisor asks you the sum of all \(S(x, y)\) for each \(0 \leq x \leq n\) and \(0 \leq y \leq m\). Help him by finding the value of \(\sum_{x=0}^{n} {\sum_{y=0}^{m} {S(x, y)}}\).
Output
Output an integer denoting the value of \(\sum_{x=0}^{n} {\sum_{y=0}^{m} {S(x, y)}}\).
Note
In the first example, the nearest telephone pole for the points \((0,0)\), \((1,0)\), \((2,0)\), \((0,1)\), \((1,1)\), and \((2,1)\) is at \((0, 0)\). While the nearest telephone pole for the points \((0, 2)\), \((1,2)\), and \((2,2)\) is at \((0, 2)\). Thus, \(\sum_{x=0}^{n} {\sum_{y=0}^{m} {S(x, y)}} = (0 + 1 + 4) + (1 + 2 + 5) + (0 + 1 + 4) = 18\).
Примеры
| № | Входные данные | Выходные данные |
|
1
|
2 2
101
000
000
|
18
|
|
2
|
5 4
10010
00000
01000
00001
00100
00010
|
36
|