
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
On an infinite checkerboard, the union of any two distinct unit squares is called a (disconnected) domino. A domino is said to be of type \((a, b)\), with \(a \leq b\) integers not both zero, if the centers of the two squares are separated by a distance of \(a\) in one orthogonal direction and \(b\) in the other. (For instance, an ordinary connected domino is of type \((0,1)\), and a domino of type \((1,2)\) contains two squares separated by a knight's move.) Each of the three pairs of squares above forms a domino of type \((1,2)\).

Two dominoes are said to be congruent if they are of the same type. A rectangle is said to be \((a, b)\)-tileable if it can be partitioned into dominoes of type \((a, b)\).

1. Let \(0 < m \leq n\) be integers. How many different (i.e., noncongruent) dominoes can be formed by choosing two squares of an \(m \times n\) array?

Present the answer in LaTex format: \boxed{Your answer}
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