
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 point \( P \) on the hyperbola with eccentricity \( \sqrt{2} \) given by \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) (\(a > 0, b > 0\)), let \( F_{1} \) and \( F_{2} \) be the two foci of the hyperbola. Given that \(\overrightarrow{P F_{1}} \cdot \overrightarrow{P F_{2}}=0\), find the ratio of the inradius \( r \) to the circumradius \( R \) of triangle \( \triangle P F_{1} F_{2} \).

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