
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
The right vertex is \(A\), \(P\) is any point on the ellipse \(C_{1}\), and \(\overrightarrow{P F_{1}} \cdot \overrightarrow{P F_{2}}\) attains its maximum value within the range \(\left[c^{2}, 3 c^{2}\right]\), where \(c=\sqrt{a^{2}-b^{2}}\).
(1) The range of values for the eccentricity \(e\) of the ellipse \(C_{1}\);
(2) Let the hyperbola \(C_{2}\) have its vertices at the foci of the ellipse \(C_{1}\), with the focuses as vertices. \(B\) is any point on the hyperbola \(C_{2}\) in the first quadrant. 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 true? If it exists, find the value of \(\lambda\); if not, provide an explanation.

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