
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 $C: y^{2}=2px\left(p \gt 0\right)$ with focus $F$ and point $A\left(2,a\right)$ on the parabola $C$, where $AF=3$.<br/>$(1)$ Find the equation of the parabola $C$ and the coordinates of the focus;<br/>$(2)$ A line $l$ passing through the focus $F$ intersects the parabola $C$ at points $M$ and $N$. If point $B\left(-1,1\right)$ satisfies $\angle MBN=90^{\circ}$, find the equation of the line $l$.

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