
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 the line $y=kx+1$ intersects the hyperbola $3x^{2}-y^{2}=1$ at point $A$ on the left branch and at point $B$ on the right branch.

(1) Find the range of real values for $k$.
(2) If a circle with diameter $AB$ passes through the origin, find the value(s) of $k$.

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