
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 a pyramid $P-ABC$ whose base $ABC$ is an equilateral triangle and whose vertex $P$ is equidistant from $A, B$, and $C$. If each side of the base $ABC$ equals 2 and the line segment joining $P$ to any vertex makes an angle $\alpha$ with the edge of the triangle, calculate the volume of the pyramid.
A) $\frac{2\tan(\alpha)}{3}$  
B) $\frac{2\sqrt{3}\tan(\alpha)}{2}$  
C) $\frac{2\sqrt{3}\tan(\alpha)}{3}$  
D) $\frac{\sqrt{3}\tan(\alpha)}{6}$

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