
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 $\triangle PQR$.
2. Find the maximum area of the triangle whose vertices are the incenters of $\triangle APB$, $\triangle BQC$, and $\triangle CRA$.

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