
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
Given an equilateral triangle $ABC$ with side length 2, points $P$, $Q$, and $R$ on sides $AB$, $BC$, and $CA$ respectively such that $\overline{PQ}$, $\overline{QR}$, and $\overline{RP}$ are parallel to $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ respectively, forming a smaller equilateral triangle $PQR$ inside $\triangle ABC$, find $PQ + QR + RP$ given that the perimeter of triangle $PQR$ equals the perimeter of trapezoid $APBR$ and the perimeter of trapezoid $BQCP$.

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