OKEA

You work for a well-known department store that uses leading technologies and employs mechanistic work — that is, robots!

The department you work in sells $$$n \cdot k$$$ items. The first item costs $$$1$$$ dollar, the second item costs $$$2$$$ dollars, and so on: $$$i$$$-th item costs $$$i$$$ dollars. The items are situated on shelves. The items form a rectangular grid: there are $$$n$$$ shelves in total, and each shelf contains exactly $$$k$$$ items. We will denote by $$$a_{i,j}$$$ the price of $$$j$$$-th item (counting from the left) on the $$$i$$$-th shelf, $$$1 \le i \le n, 1 \le j \le k$$$.

Occasionally robots get curious and ponder on the following question: what is the mean price (arithmetic average) of items $$$a_{i,l}, a_{i,l+1}, \ldots, a_{i,r}$$$ for some shelf $$$i$$$ and indices $$$l \le r$$$? Unfortunately, the old robots can only work with whole numbers. If the mean price turns out not to be an integer, they break down.

You care about robots' welfare. You want to arrange the items in such a way that the robots cannot theoretically break. Formally, you want to choose such a two-dimensional array $$$a$$$ that:

• Every number from $$$1$$$ to $$$n \cdot k$$$ (inclusively) occurs exactly once.
• For each $$$i, l, r$$$, the mean price of items from $$$l$$$ to $$$r$$$ on $$$i$$$-th shelf is an integer.

Find out if such an arrangement is possible, and if it is, give any example of such arrangement.