
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 a hyperbola $C$ centered at the origin with its right focus at $(2,0)$ and a real axis length of $2 \sqrt{3}$.

(1) Find the equation of the hyperbola $C$.
(2) If the line $l: y = kx + \sqrt{2}$ intersects the left branch of the hyperbola $C$ at points $A$ and $B$, find the range of $k$.

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