
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
The parabola $C$ has its vertex at the origin of coordinates, its focus on the $x$-axis, and passes through the point $P(2,2)$.
$(1)$ Find the standard equation of the parabola and the coordinates of the focus;
$(2)$ The line $l: x-y-1=0$ intersects the parabola $C$ at points $M$ and $N$, find $|MN|$.

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