
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
A square is given. Inscribe an equilateral triangle in the square such that one of its vertices coincides with one of the vertices of the square. Then, inscribe a square in this triangle such that two of the square's vertices lie on one side of the triangle, with the third and fourth vertices on the other sides of the triangle. Repeat this process indefinitely. What is the sum of the areas of all these squares if the side length of the original square is $a$?

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