
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
A hyperbola $C$ has its center at the origin, right focus at $F\left( \frac{2 \sqrt{3}}{3},0\right)$, and asymptote lines with equations $y=± \sqrt{3}x$.

(I) Find the equation of hyperbola $C$;

(II) Suppose line $l:y=kx+1$ intersects hyperbola $C$ at points $A$ and $B$. If a circle passing through the origin has chord $AB$ as its diameter, find the value of $k$.

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