
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 an ellipse $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with left and right foci $F_{1}$ and $F_{2}$, and focal length $2\sqrt{3}$, a line $m$ passing through $F_{1}$ intersects the ellipse $C$ at points $A$ and $B$, and the perimeter of $\triangle ABF_{2}$ is $8$.  
$(1)$ Find the equation of the ellipse $C$;  
$(2)$ If a moving line $n$ passing through point $G(1,0)$ intersects the ellipse $C$ at points $M$ and $N$, and the equation of line $l$ is $x=4$. Line $l$ is perpendicular to $MP$ at point $P$ and perpendicular to $NQ$ at point $Q$. Let the areas of $\triangle GPQ$, $\triangle GPM$, and $\triangle GQN$ be $S$, $S_{1}$, and $S_{2}$, respectively. Determine if there exists a real number $\lambda$ such that $λ\sqrt{S_{1}⋅S_{2}}-S=0$ holds. If it exists, find the value of $\lambda$; if not, explain the reason.

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