
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 that the distance from point $M(5, m)$ on the parabola $C: y^{2} = 2px (p > 0)$ to the focus $F$ is $6$.  
$(1)$ Find the equation of the parabola $C$.  
$(2)$ A line $l$ passing through point $P(2, -1)$ intersects the parabola $C$ at points $A$ and $B$, where point $P$ is the midpoint of segment $AB$. Find the equation of the line $l$.

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