
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 $xOy$, 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 curve C be the part of the ellipse in the first quadrant. A moving point P is on C, and the tangent line to C at point P intersects the x-axis and y-axis at A and B, respectively, with the vector property $\overrightarrow{OM} = \overrightarrow{OA} + \overrightarrow{OB}$. Find:
(I) The equation of the trajectory of point M;
(II) The minimum value of $|\overrightarrow{OM}|$.

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