
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
In the Cartesian coordinate system $xOy$, given the line $l: x-y-2=0$ and the parabola $C: y^2=2px (p > 0)$.

$(1)$ If line $l$ passes through the focus of parabola $C$, find the equation of parabola $C$;

$(2)$ When $p=1$, if there exist two distinct points $P$ and $Q$ on parabola $C$ that are symmetric about line $l$, find the coordinates of the midpoint $M$ of segment $PQ$.

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