
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
The center of the hyperbola $C$ is at the origin, and its right focus is $F\left( \frac{2\sqrt{3}}{3}, 0\right)$, with asymptote equations $y=\pm \sqrt{3}x$.
- (I) Find the equation of the hyperbola $C$;
- (II) Let the line $l: y=kx+1$ intersect the hyperbola $C$ at points $A$ and $B$. What value of $k$ makes the circle with diameter $AB$ pass through the origin?

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