
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 a hyperbola
\[ C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \quad (a>0, b>0) \]
where \( F_{1} \) and \( F_{2} \) are the left and right foci of the hyperbola \( C \), respectively, and \( P \) is a point on the right branch of the hyperbola \( C \) such that \( \angle F_{1} P F_{2} = \frac{\pi}{3} \). Suppose the area of the triangle \( \triangle F_{1} P F_{2} \) is \( 3 \sqrt{3} a^{2} \).

1. Find the eccentricity \( e \) of the hyperbola \( C \).
2. Let \( A \) be the left vertex of the hyperbola \( C \), and \( Q \) be any point on the hyperbola \( C \) in the first quadrant. Does there exist a constant \( \lambda (\lambda > 0) \) such that \( \angle Q F_{2} A = \lambda \angle Q A F_{2} \) always holds? If it exists, find the value of \( \lambda \); if it does not, explain the reason.

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