
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 hyperbola $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ with $a, b > 0$. Points $F_{1}$ and $F_{2}$ are the left and right foci of $C$, respectively. Point $P$ is on the right branch of $C$ and satisfies $\angle F_{1} \mathrm{PF}_{2}=\frac{\pi}{3}$. Additionally, the area of triangle $\mathrm{F}_{1} \mathrm{PF}_{2}$ is $3 \sqrt{3} \mathrm{a}^{2}$.

1. Find the eccentricity $e$ of $C$.
2. Let $A$ be the left vertex of $C$, and $Q$ be any point on $C$ in the first quadrant. Determine whether there exists a constant $\lambda(\lambda>0)$ such that $\angle \mathrm{QF}_{2}A = \lambda \angle \mathrm{QAF}_{2}$ 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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