
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 the parabola $C$: $y^{2}=2px$, and point $P(1,2)$ is on the parabola.  
$(1)$ Find the value of $p$  
$(2)$ Line $l$ passes through the focus and intersects the parabola at points $A$ and $B$. If $|AB|=10$, find the equation of line $l$.

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