
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
A moving point $M$ is at the same distance from the line $l: x = -1$ as it is from the fixed point $F(1, 0)$.

$(1)$ Find the equation of the trajectory $C$ of point $M$;

$(2)$ Suppose a line $l_{1}$ passing through point $F$ with a slope of $k$ intersects the curve $C$ at two points $A$ and $B$, and $|AB| = 6$, find the equation of $l_{1}$.

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