
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
An equilateral triangle $ABP$, with side $AB$ of length $3$ inches, is placed inside a square $AXYZ$ with a side of length $6$ inches so that vertex $B$ is on side $AX$. The triangle is rotated clockwise about $B$, then $P$, and so on, along the sides of the square until vertex $P$ returns to its original position. There is an additional condition that the triangle must rotate around each corner of the square twice before moving to the next corner. Calculate the length of the path in inches traversed by vertex $P$.
A) $12\pi$
B) $16\pi$
C) $20\pi$
D) $24\pi$
E) $28\pi$

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