
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 points \( A \) and \( B \) are on an ellipse \(\Gamma\) with eccentricity \( e \):
\[ 
\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \quad (a > b > 0) 
\]
\( F_{1} \) and \( F_{2} \) are the two foci of the ellipse \(\Gamma\), and \( O \) is the origin such that \( OA \perp OB \). Additionally,
\[ 
\overrightarrow{A F_{1}} \cdot \overrightarrow{A F_{2}} + \overrightarrow{B F_{1}} \cdot \overrightarrow{B F_{2}} = 0 
\]
Determine the range of values for \( e \).

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