
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

Let the triangle $\Delta ABC$ be equilateral.

1. How does the perimeter and area of the shape $APMQ$ change if point $M$ moves along the side $BC$? Assume $BC = a$.

2. Let $M$ be the midpoint of $BC$, and rotate the shape $APMQ$ around $BC$. Calculate the volume of the resulting solid of revolution.

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