
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
A right triangle $ABP$, with $AB$ as the hypotenuse and $AP$ and $BP$ as the legs, is placed inside a square $AXYZ$ with side length $8$ inches such that vertex $B$ is on side $AX$. Side $AP$ is $4$ inches and side $BP$ is $4\sqrt{3}$ inches. The triangle is rotated clockwise about $B$, then $P$, and so on along the sides of the square until $P$ returns to its original position. Calculate the length of the path in inches traversed by vertex $P$.
A) $8\pi$
B) $12\pi$
C) $16\pi$
D) $20\pi$

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