Usually, smartphones can be locked by either a password, fingerprint sensing, or facial recognition. There's also another commonly used method: Pattern Lock.

Pattern Lock allows you to lock and unlock your device by drawing a pattern onto the screen. The pattern is drawn by connecting a series of dots on a grid by a polygonal path. The grid of dots on the lock screen has $$$n$$$ rows and $$$m$$$ columns. The rows have equal spacing between each other, and so do the columns. We denote the dots in the $$$x$$$-th row and $$$y$$$-th column by $$$(x,y)$$$. And we use a sequence of dots to denote the pattern, i.e. the polygonal path. For example, the sequence $$$\{(1,1),(2,3),(3,2),(2,2)\}$$$ can denote the pattern shown in the picture below.

Let's denote a pattern with $$$k$$$ dots as $$$\{A_1,A_2,\dots,A_k\}$$$. A valid pattern to lock a smartphone should meet the following conditions:

• Each dot is visited no more than once. That is, for each $$$1 \le i \lt j \le k$$$, $$$A_i \neq A_j$$$.
• For each $$$1 \le i \lt k$$$, the segment connecting $$$A_i$$$ and $$$A_{i+1}$$$ cannot pass through other dots. For example, $$$A_i=(1,1),A_{i+1}=(3,3)$$$ is invalid because the segment passes through $$$(2,2)$$$.

Little Rabbit wants his smartphone to be as secure as possible. Therefore, he needs a strong pattern to lock his device. A strong pattern is a valid pattern that meets some extra conditions:

• Each dot is visited exactly once. That is, $$$k=n \times m$$$.
• For each $$$1 \lt i \lt k$$$, the angle formed by segment $$$A_iA_{i-1}$$$ and segment $$$A_iA_{i+1}$$$ must be an acute angle (less than $$$90^\circ$$$).

Can you construct a strong pattern for him?