
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 $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\left(a \gt b \gt 0\right)$, with the left vertex $A$, the top vertex $B$, and $|AB|=\sqrt{7}$. A line $l$ is drawn passing through the right focus $F$, and when the line $l$ passes through point $B$, the slope is $-\sqrt{3}$.  
$(1)$ Find the equation of $C$;  
$(2)$ If $l$ intersects $C$ at points $P$ and $Q$, and there exists a point $M$ on $l$ such that $\overrightarrow{QM}=\overrightarrow{FP}$, is there a pair of fixed points in the plane such that the sum of the distances from point $M$ to these two fixed points is constant? If so, find these two fixed points and the constant value; if not, explain the reason.

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