Maximize the Minimum Distance

Обмеження: 2 сек., 512 МіБ

This problem is different from the problem "Minimize the Maximum Distance".

There are \(n \cdot m\) points on a two-dimensional plane located in \(n\) rows and \(m\) columns.

Distances between adjacent rows and columns are all equal to \(1\).

You want to paint all the points in some colors. We denote all colors you can use with integers \(1, 2, \dots, 10^5\). The number of points for each color you use must be two or three.

Your goal is to maximize the minimum Euclidean distance between points of the same color. You need not only to find the value of the sought distance but also to find the actual colors of the points.

Вхідні дані

The single line contains two integers \(n\) and \(m\) – the number of rows and columns, respectively.

Вихідні дані

In the first line, print an integer number – the square of the maximum possible minimum distance between points of the same color.

In the following \(n\) lines, print \(m\) integers \(c_{i j}\) that denote the colors of the points. These numbers must satisfy \(1 \le c_{i j} \le 10^5\). Each of these numbers must occur exactly two or three times.

If there are multiple answers, any of them will be accepted.

Обмеження

\(1 \le n, m\),

\(2 \le n \cdot m \le 10^5\).

Приклади

Вхідні дані (stdin) копіювати	Вихідні дані (stdout) копіювати
2 3	2 4 7 4 7 4 7

Вхідні дані (stdin) копіювати	Вихідні дані (stdout) копіювати
4 3	5 1 2 3 3 4 5 6 1 2 4 5 6

Вхідні дані (stdin) копіювати	Вихідні дані (stdout) копіювати
1 5	4 4 7 4 7 4

Примітки