
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
Right-angled triangle $ABP$, where angle $B$ is $90^\circ$ and sides $AB = 2$ inches and $BP = 4$ inches, is placed inside square $AXYZ$ with a side of length $6$ inches so that $B$ is on side $AX$. The triangle is rotated clockwise about $B$, then $P$, then $A$, and so on along the sides of the square until $P$ returns to its original position. Calculate the total length of the path in inches that vertex $P$ traverses.
A) $18\pi$
B) $24\pi$
C) $32\pi$
D) $40\pi$

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