
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 $C$: $\frac{x^2}{3} - y^2 = 1$.  
$(1)$ If point $P$ lies on the curve $C$, and points $A$ and $B$ lie on the asymptotes $l_{1}$ and $l_{2}$ of the hyperbola $C$, with point $A$ in the first quadrant and point $B$ in the fourth quadrant, and if $\overrightarrow{AP} = \lambda \overrightarrow{PB}$, where $\lambda \in [\frac{1}{3}, 2]$, find the maximum area of triangle $\triangle AOB$.  
$(2)$ Let the left and right foci of the hyperbola $C$ be $F_{1}$ and $F_{2}$, respectively. A line $l$ passing through the left focus $F_{1}$ intersects the left branch of the hyperbola at points $G$ and $Q$. Find the range of values for the perimeter of triangle $\triangle GQF_{2}$.

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