
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 $E$: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\left(a \gt b \gt 0\right)$, the area of the quadrilateral formed by connecting the four vertices of ellipse $E$ is $4\sqrt{3}$.<br/>$(1)$ If $a=2$, find the standard equation of ellipse $E$;<br/>$(2)$ Let $G$ be the parabola with the right vertex of ellipse $E$ as the focus. If the shortest distance from a moving point $M$ on $G$ to the point $H\left(10,0\right)$ is $4\sqrt{6}$, find the value of $a$;<br/>$(3)$ When $a=2$, let point $F$ be the right focus of ellipse $E$, $A\left(-2,0\right)$, and line $l$ intersect $E$ at points $P$ and $Q$ (neither coinciding with point $A$). The slopes of lines $l$, $AP$, and $AQ$ are $k$, $k_{1}$, and $k_{2}$, respectively. If $kk_{1}+kk_{2}+3=0$, find the perimeter of $\triangle FPQ$.

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