
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 Cartesian coordinate system $xOy$, the right focus of the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ is $F(\sqrt{3},0)$. The area of the triangle formed by the upper and lower vertices of $C$ and $F$ is $\sqrt{3}$.
$(1)$ Find the equation of $C$;
$(2)$ Given that the line $l$ passing through point $F$ intersects $C$ at points $A$ and $B$, is there a point $Q$ on $C$ such that $\overrightarrow{OA} + \overrightarrow{OB} = \overrightarrow{OQ}$? If yes, find the equation of $l$. If not, explain why.

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