
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 a parabola $C$ whose focus is on the $x$-axis and vertex at the origin, and it passes through the point $P(2,1)$. A line $l$ passing through the point $(2,0)$ intersects the parabola $C$ at points $A$ and $B$. Let $M$ be the midpoint of the line segment $AB$, and a perpendicular line to the $y$-axis through point $M$ intersects $C$ at point $N$.
$(1)$ Find the equation of the parabola $C$;
$(2)$ Determine whether there exists a line $l$ such that the circle with diameter $AB$ passes through point $N$. If it exists, find the equation of line $l$; if not, explain why.

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