
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
We have an equilateral triangle with circumradius  $1$ . We extend its sides. Determine the point  $P$  inside the triangle such that the total lengths of the sides (extended), which lies inside the circle with center  $P$  and radius  $1$ , is maximum.
(The total distance of the point P from the sides of an equilateral triangle is fixed )

*Proposed by Erfan Salavati*

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