
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 \( C: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \) (where \( a, b > 0 \)), with left and right foci denoted as \( F_{1} \) and \( F_{2} \) respectively, and given that the hyperbola intersects with the circle \( x^{2}+y^{2}=r^{2} \) (where \( r > 0 \)) at a point \( P \). If the maximum value of \( \frac{\left| P F_{1} \right| + \left| P F_{2} \right|}{r} \) is \( 4 \sqrt{2} \), determine the eccentricity of the hyperbola \( C \).

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