
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
In a Cartesian coordinate system, \( F_{1} \) and \( F_{2} \) are the foci of a hyperbola \( \Gamma: \frac{x^{2}}{3} - y^{2} = 1 \). A point \( P \) on \( \Gamma \) satisfies \( \overrightarrow{P F_{1}} \cdot \overrightarrow{P F_{2}} = 1 \). Find the sum of the distances from point \( P \) to the two asymptotes of \( \Gamma \).

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