
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-angled isosceles triangle $ABP$ with $AB = BP = 4$ inches is placed inside a square $AXYZ$ with side length $8$ inches so that point $B$ is on side $AX$ of the square. The triangle is rotated clockwise about the midpoint of each side ($AB$, then $BP$, etc.), continuing this process 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) $12\pi\sqrt{2}$ inches
B) $24\pi$ inches
C) $8\pi\sqrt{2}$ inches
D) $16\pi$ inches

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