
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 an ellipse $E$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1$ $(a > b > 0)$, the four vertices of the ellipse form a quadrilateral with an area of $2 \sqrt{3}$. One of the internal angles of the quadrilateral is $60^{\circ}$.
(I) Find the equation of the ellipse $E$;
(II) Line $l$ intersects with ellipse $E$ at two distinct points $A$ and $B$. The midpoint of segment $AB$ is $C$, and $O$ is the coordinate origin. If the area of $\triangle OAB$ is $\frac{\sqrt{3}}{2}$, find the minimum value of $|OC|$.

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