
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
In the plane rectangular coordinate system $xOy$, the ellipse $C$: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\left(a \gt b \gt 0\right)$ has the upper and lower vertices $A$ and $B$ respectively. Point $F$ is the right focus of the ellipse, and $\overrightarrow{FA}\cdot \overrightarrow{FB}=2$, $S_{\triangle ABF}=\sqrt{3}$. <br/> $(1)$ Find the equation of the ellipse $C$; <br/> $(2)$ If the line $l$ with slope $k\left(k\in R\right)$ intersects the ellipse $C$ at points $P$ and $Q$, with $O$ as the origin, and the product of the slopes of lines $OP$ and $OQ$ is $-\frac{1}{4}$, try to find out if the area of $\triangle OPQ$ is a constant value, and explain the reason.

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