
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
In the Cartesian coordinate system $xoy$, the sum of the distances from point $P$ to the two points $(0, -\sqrt{3})$ and $(0, \sqrt{3})$ equals $4$. Let the trajectory of point $P$ be $C$.
$(1)$ Find the equation of the trajectory $C$;
$(2)$ Suppose the line $y= \frac{1}{2}x$ intersects $C$ at points $A$ and $B$, find the length of chord $AB$.

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