
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 hyperbola $C$ with its center at the origin has its right focus at $F\left( \frac{\sqrt{6}}{2}, 0\right)$, and its asymptotes are given by the equations $y = \pm \sqrt{2}x$.  
$(I)$ Find the equation of the hyperbola $C$;  
$(II)$ A line $l: y = kx - 1$ intersects the hyperbola $C$ at points $P$ and $Q$. Investigate whether there exists a circle with diameter $PQ$ that passes through the origin. If it exists, find the value of $k$; if not, explain why.

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