
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 a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \left( a > 0, b > 0 \right)$, the left and right foci are $F_1$ and $F_2$, respectively. The line passing through point $F_1$ and perpendicular to the x-axis intersects the left branch of the hyperbola at points $A$ and $B$. The lines $AF_2$ and $BF_2$ intersect the y-axis at points $P$ and $Q$, respectively. If the perimeter of $\triangle PQF_2$ is $12$, find the eccentricity of the hyperbola when $ab$ is at its maximum value.

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