
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 $E$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, its left and right foci are $F_1$ and $F_2$ respectively, ($|F_1F_2| = 6$). $P$ is a point on the right branch of $E$, and the line segment $PF_1$ intersects the $y$-axis at point $A$. The incircle of $\triangle PAF_2$ intersects the side $AF_2$ at point $Q$. If $|AQ| = \sqrt{3}$, calculate the eccentricity of $E$.

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