
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 circle $C$ with its center at point $C(0,3)$ and point $D(-\sqrt{3},2)$ on circle $C$, a line $l$ passes through point $A(-1,0)$ and intersects circle $C$ at points $P$ and $Q$, with point $M$ being the midpoint of segment $PQ$.  
$(1)$ Find the equation of circle $C$.  
$(2)$ If $|AM|=3$, find the equation of line $l$.

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