
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
In a square $WXYZ$ with a side length of $6$ inches, an equilateral triangle $ABC$ with each side of $3$ inches is placed such that vertex $C$ is on the side $WZ$. The triangle rotates clockwise about vertex $C$, then $A$, then $B$, consecutively around the edges of the square until vertex $C$ returns to its original position. Determine the length of the path in inches traversed by vertex $A$.

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