
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
A quartet is defined as a group of four cells on graph paper, with their centers located at the vertices of a rectangle whose sides are parallel to the grid lines. (For instance, the picture shows three quartets.)

What is the maximum number of quartets that can be arranged in:
a) a $5 \times 5$ square;

b) an $m \times n$ rectangle of cells?

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