
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 the point $\left(2,t\right)$ on the parabola $C:y^{2}=2px\left(p \gt 0\right)$ to the focus $F$ is $4$. <br/>$(1)$ Find the standard equation of the parabola $C$; <br/>$(2)$ The line $y=kx+1$ intersects the parabola $C$ at two distinct points $A$ and $B$. If $\overrightarrow{FA}•\overrightarrow{FB}=4$, find the equation of the line $l$.

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