
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 $f(x)=ax-\frac{1}{x}$, $g(x)=\ln x$, $x > 0$, where $a\in R$ is a constant.
(I) Find the equation of the tangent line to the curve $y=g(x)$ at the point $P(1,g(1))$;
(II) Let $F(x)=f(x)-g(x)$, discuss the monotonicity of the function $F(x)$.

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