
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 \( C: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \) with \( a > 0 \) and \( b > 0 \), the left and right foci are \( F_{1} \) and \( F_{2} \). A circle with diameter \( F_{1}F_{2} \) intersects the hyperbola \( C \) at point \( P \) in the second quadrant. If the eccentricity of the hyperbola is 5, find the value of \( \cos \angle P F_{2} F_{1} \).

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