
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 $C$: $\frac{x^2}{2}-\frac{y^2}{b^2}=1\left(b \gt 0\right)$, where the distance from a focus $F$ to the asymptote is $\sqrt{2}$. <br/>$(1)$ Find the equation of the hyperbola $C$; <br/>$(2)$ A line $l$ passing through the point $\left(2,0\right)$ intersects the right branch of the hyperbola $C$ at points $A$ and $B$. Does there exist a point $N$ on the $x$-axis such that $\overrightarrow{NA}•\overrightarrow{NB}$ is a constant value? If yes, find the coordinates of point $N$ and the constant value; if not, explain why.

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