
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
In an equilateral triangle \( \triangle ABC \) with unit area, external equilateral triangles \( \triangle APB \), \( \triangle BQC \), and \( \triangle CRA \) are constructed such that \(\angle APB = \angle BQC = \angle CRA = 60^\circ\). 
1. Find the maximum area of \( \triangle PQR \).
2. Find the maximum area of the triangle formed by the incenter of \( \triangle APB \), the incenter of \( \triangle BQC \), and the incenter of \( \triangle CRA \).

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