
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
Let \( m, n \) be natural numbers. Consider a square grid of \( (2m+1) \times (2n+1) \) points in the plane. A set of rectangles is called good if the following holds:

(a) For each of the rectangles, the four corner points lie on grid points and the sides are parallel to the grid lines.

(b) No two of the rectangles have a common corner point.

Determine the largest possible value of the sum of the areas of all rectangles in a good set.

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