
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 isosceles triangle $ABP$ with sides $AB = AP = 3$ inches and $BP = 4$ inches is placed inside a square $AXYZ$ with a side length of $8$ inches, such that $B$ is on side $AX$. 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 total path length in inches traversed by vertex $P$.
A) $\frac{24\pi}{3}$
B) $\frac{28\pi}{3}$
C) $\frac{32\pi}{3}$
D) $\frac{36\pi}{3}$

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