
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 a circle $C$: $x^{2}+y^{2}+2x+a=0$ has two points symmetric about the line $l$: $mx+y+1=0$.

$(1)$ Find the value of the real number $m$;

$(2)$ If the line $l$ intersects the circle $C$ at points $A$ and $B$, and $\overrightarrow{OA} \cdot \overrightarrow{OB}=-3$ ($O$ is the origin), find the equation of the circle $C$.

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