
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 two fixed points $A(0,3)$ and $O(0,0)$ on the coordinate plane, and a moving point $M(x,y)$ that satisfies $|MA|=2|OM|$.

(1) Find the trajectory equation of point $M$ and describe the shape of the trajectory.

(2) Denote the trajectory in (1) as $C$. A line $l$ passing through point $N(-1,3)$ intercepts a line segment of length $2\sqrt{3}$ on $C$. Find the equation of line $l$.

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