
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
Let \( F_1 \) and \( F_2 \) be the left and right foci of the ellipse \( C \):

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a > b > 0) \]

If there exists a point \( P \) on the ellipse \( C \) such that \( P F_1 \) is perpendicular to \( P F_2 \), then determine the range of the eccentricity \( e \) of the ellipse.

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