
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 that the asymptotic line equation of the hyperbola $C$ is $y=\pm\sqrt{3}x$, point $P(2,3)$ lies on the hyperbola $C$. Line $l:y=kx+m$ intersects the hyperbola at points $A$ and $B$, with slopes of $k_{1}$ and $k_{2}$, respectively.  
$(1)$ Find the standard equation of the hyperbola $C$.  
$(2)$ Determine if there exists a constant $k$ such that $k_{1}k_{2}$ is a constant value $\lambda$. If it exists, find the values of the constant $k$ and $\lambda$. If it does not exist, provide a reason.

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