
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 an ellipse \( C_{1}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0) \) with foci \( F_{1} \) and \( F_{2} \) on the left and right, respectively, and right vertex \( A \). Let \( P \) be any point on the ellipse \( C_{1} \), and the range for the maximum value of \( \overrightarrow{P F_{1}} \cdot \overrightarrow{P F_{2}} \) is \([c^{2}, 3 c^{2}]\), where \( c = \sqrt{a^{2} - b^{2}} \).

1. Find the range of the eccentricity \( \boldsymbol{e} \) of the ellipse \( C_{1} \):
    
2. Given a hyperbola \( C_{2} \) with the foci of ellipse \( C_{1} \) as its vertices, and \( B \) being any point on \( C_{2} \) in the first quadrant, when the eccentricity \( e \) takes the minimum value, is there a constant \( \lambda (\lambda > 0) \) such that \( \angle B A F_{1} = \lambda \angle B F_{1} A \) always holds? If it exists, find the value of \( \lambda \); if it does not exist, explain why.

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