
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 common chord of two intersecting circles is equal to \( a \) and serves as the side of a regular inscribed triangle for one circle and as the side of an inscribed square for the other. Determine the distance between the centers of the circles (consider two possible cases).

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