
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 AB be two points on the parabola C: x²=2py (p>0), the x-coordinate of the midpoint of AB is 2, and the slope of the line AB is 1.
(1) Find the equation of the parabola C;
(2) Line l: x=t (t≠0) intersects the x-axis at point M and intersects the parabola C: x²=2py (p>0) at point P. The symmetric point of M about P is N. Connect ON and extend it to intersect C at point H. Does line MH have other common points with C besides H? Please explain the reason.

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