
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 $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}} (a > b > 0)$ passing through the point $(1, \frac{\sqrt{3}}{2})$, its two minor axis endpoints and the right focus form an equilateral triangle. Point $A$ moves on the ellipse $C$, and point $B$ moves on the line $l$: $y = m (m > 0)$, such that $\angle AOB = 90^{\circ}$ (where $O$ is the origin).
(I) Find the equation of the ellipse $C$:
(II) If the distance from point $O$ to line $AB$ is a constant value, find the value of $m$ and the minimum value of $|AB|$.

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