
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 $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0,b > 0$), with the left and right foci denoted as $F_1$ and $F_2$ respectively, a point $P$ is on the right branch of the hyperbola, and $|PF_1| = 4|PF_2$. Find the maximum possible value for the eccentricity $e$ of this hyperbola.

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