
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 the equation of a circle: $x^{2}+y^{2}=4$.

(I) A line $L$ passes through point $P(1,2)$ and intersects circle $C$ at points $A$ and $B$. If $|AB|=2\sqrt{3}$, find the equation of line $L$.

(II) A line $m$ parallel to the $X$ axis passes through a moving point $M$ on circle $C$. Let the intersection point of $m$ and the $y$ axis be $N$. If vector $\overrightarrow{OQ}= \overrightarrow{OM}+ \overrightarrow{ON}$ (where $O$ is the origin), find the trajectory equation of the moving point $Q$.

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