
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 that point \( P \) is on the hyperbola with eccentricity \( \sqrt{2} \):
\[ 
\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \quad (a > 0, b > 0) 
\]
where \( F_{1} \) and \( F_{2} \) are the two foci of the hyperbola, and \(\overrightarrow{P F_{1}} \cdot \overrightarrow{P F_{2}} = 0\). Find the ratio of the inradius \( r \) to the circumradius \( R \) of triangle \( P F_{1} F_{2} \).

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