
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
Let point P be a common focus $F_1(-2, 0)$, $F_2(2, 0)$ of an ellipse and a hyperbola, and $\cos\angle F_1PF_2 = \frac{3}{5}$. It is known that the length of the major axis of the ellipse is four times the length of the real axis of the hyperbola. Determine the eccentricity of the hyperbola.

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