
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 $C:y^{2}=2px\left(p\neq 0\right)$.  
$(1)$ If a point $M\left(1,t\right)$ on $C$ is at a distance of $3$ from its focus, find the equation of $C$;  
$(2)$ If $p=2$, a line $l$ with a slope of $2$ intersects $C$ at two points, intersects the positive $x$-axis at point $M$, and $O$ is the origin with $\overrightarrow{OA}•\overrightarrow{OB}=0$, find the coordinates of point $M$.

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