
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
Let the parabola $C: y^2 = 2px$ ($p>0$) have its focus at $F$, and let $A$ be the point on the parabola with an x-coordinate of 4, and $|AF|=5$.  
(1) Find the equation of the parabola $C$;  
(2) Let the line $l$ pass through point $F$ with a slope of 1 and intersect the parabola $C$ at points $M$ and $N$, with $O$ being the origin. Find the area of $\triangle OMN$.

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