
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 right triangle with integer leg lengths. Define the triangle as "cool" if the area of the triangle in square units is equal to three times the sum of the lengths of its legs. Find the sum of all distinct possible areas of such cool right triangles.

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