
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 plane, a moving point $M(x,y)$ is on the left side of the $y$-axis, and the difference between the distance from point $M$ to a fixed point $F(-1,0)$ and the distance to the $y$-axis is $1$.  
$(1)$ Find the equation of the trajectory $C$ of the moving point $M$;  

$(2)$ If a line $l$ passing through point $P(-3,-2)$ intersects curve $C$ at points $A$ and $B$, and point $P$ is exactly the midpoint of $AB$, find the length of segment $AB$.

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