
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 $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (with $a>0, b>0$), let its left and right foci be $F_1$ and $F_2$ respectively. Suppose point $P$ lies on the hyperbola and the segment $PF_2$ is perpendicular to the x-axis. If $|F_1F_2| = 12$ and $|PF_2| = 5$, determine the eccentricity of the hyperbola.

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