
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. More 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-self-intersecting 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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