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

(1) Find the range of the eccentricity \( e \) of the ellipse \( C_{1} \).

(2) Let the hyperbola \( C_{2} \) have the same foci as the ellipse \( C_{1} \) with the foci as vertices. Point \( B \) is any point in the first quadrant of \( C_{2} \). When \( e \) takes its 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 such a value of \(\lambda\) exists, find \(\lambda\); if not, provide a reason.

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