
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 $C$: $x^{2}=2py(p > 0)$ with focus $F$, a line $l$ passes through point $F$ and intersects the parabola $C$ at points $A$ and $B$. A circle $M$ with diameter $AB$ is tangent to the line $y=-1$ at point $N$.

$(1)$ Find the equation of $C$.

$(2)$ If the circle $M$ is tangent to the line $x=-\frac{3}{2}$ at point $Q$, find the equation of the line $l$ and the equation of the circle $M$.

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