
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 equilateral triangle $ABP$ with side $AB$ of length $3$ inches is placed inside a square $AXYZ$ with a side of length $6$ inches so that $B$ is on side $AX$. The triangle is rotated clockwise about $B$. After a full rotation about $B$, it is rotated about $P$, continuing along the sides of the square until $P$ returns to its original position. Each rotation about a vertex covers an angle of $60^\circ$. Calculate the total length of the path in inches traversed by vertex $P$.
A) $8\pi$
B) $10\pi$
C) $12\pi$
D) $14\pi$

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