
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
Find the smallest positive real constant  $a$ , such that for any three points  $A,B,C$  on the unit circle, there exists an equilateral triangle  $PQR$  with side length  $a$  such that all of  $A,B,C$  lie on the interior or boundary of  $\triangle PQR$ .

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