
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
Consider a  $100\times 100$  square unit lattice  $\textbf{L}$  (hence  $\textbf{L}$  has  $10000$  points). Suppose  $\mathcal{F}$  is a set of polygons such that all vertices of polygons in  $\mathcal{F}$  lie in  $\textbf{L}$  and every point in  $\textbf{L}$  is the vertex of exactly one polygon in  $\mathcal{F}.$  Find the maximum possible sum of the areas of the polygons in  $\mathcal{F}.$ *Michael Ren and Ankan Bhattacharya, USA*

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