
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 function $f(x)=a(x-\dfrac{1}{x})-b\ln x(a,b\in \mathbb{R})$, $g(x)=x^{2}$.

(1) If $a=1$, the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$ is perpendicular to the $y$-axis, find the value of $b$;

(2) If $b=2$, investigate whether the graphs of the functions $f(x)$ and $g(x)$ have a common tangent line at their intersection point(s). If such a tangent line exists, study the number of possible values for $a$; if not, please explain why.

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