
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
Line $l$ passes through the focus $F$ of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) and is perpendicular to the real axis. $A$ and $B$ are two vertices of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. If there exists a point $P$ on line $l$ such that $\angle APB = 60^\circ$, find the maximum value of the eccentricity of the hyperbola.

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