
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 $C:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)$ has an eccentricity of $\sqrt{3}$. The circle $O:x^{2}+y^{2}=2$ intersects the positive $x$-axis at point $A$. The chord cut by the tangent of circle $O$ at point $A$ on the hyperbola $C$ has a length of $2\sqrt{2}$.  
$(1)$ Find the equation of the hyperbola $C$;  
$(2)$ Let $P$ be any point on circle $O$, and the tangent at point $P$ intersects the hyperbola $C$ at points $M$ and $N$. Determine whether $|PM|\cdot |PN|$ is a constant value. If it is a constant value, find the value. If it is not a constant value, explain the reason.

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