
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 an equilateral triangular grid \( G \) with 20 points on a side, where each row consists of points spaced 1 unit apart. Specifically, there is a single point in the first row, two points in the second row, ..., and 20 points in the last row, for a total of 210 points. Let \( S \) be a closed non-selfintersecting polygon which has 210 vertices, using each point in \( G \) exactly once. Find the sum of all possible values of the area of \( S \).

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