
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 plane $(xOy)$, there is an ellipse with foci at $F\_1(0,-\sqrt{3})$ and $F\_2(0,\sqrt{3})$, and an eccentricity of $\frac{\sqrt{3}}{2}$. Let the portion of the ellipse in the first quadrant be denoted as curve $C$, and let point $P$ move along $C$. The tangent line to $C$ at point $P$ intersects the $x$- and $y$-axes at points $A$ and $B$ respectively, and $\overrightarrow{OM}=\overrightarrow{OA}+\overrightarrow{OB}$. Find:
(I) the trajectory equation of point $M$;
(II) the minimum value of $|\overrightarrow{OM}|$.

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