
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 focus of the parabola $y^{2}=2px\left(p \gt 0\right)$ is $F$, and point $A(2$,$y_{0})$ is a point on the parabola, with $|AF|=4$.  
$(1)$ Find the equation of the parabola.  
$(2)$ A line $l:y=x+m$ that does not pass through the origin intersects the parabola at two distinct points $P$ and $Q$. If $OP\bot OQ$, find the value of $m$.

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