
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
Consider an equilateral triangle $ABP$ with a side length of $3$ inches placed inside a square $AXYZ$ with a side length of $6$ inches such that vertex $B$ is on side $AX$. The triangle first translates such that $AB$ slides along diagonal $AX$ for half its length while $P$ also rotates about $B$. After reaching the midpoint of $AX$, $P$ then continues rotating around the midpoint of side $XY$, eventually returning to its original position. Compute the total path in inches traversed by vertex $P$.
A) $9\pi$ inches
B) $12\pi$ inches
C) $15\pi$ inches
D) $18\pi$ inches

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