
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
Given the hyperbola C: x² - $\frac{y^{2}}{3}$ = 1, with left and right foci $F_1$ and $F_2$, and left and right vertices A and B, respectively. A line passing through point $F_1$ intersects the right branch of hyperbola C at point P, such that |$\overrightarrow{AP}$|cos<>$\overrightarrow{AP}$, $\overrightarrow{AF_{2}}$> = |$\overrightarrow{AF_{2}}|$. Calculate the area of the circumcircle of ΔABP.

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