
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 functions $f(x) = \ln x$ and $g(x) = ax + \frac{b}{x}$, if the intersection point of the graph of $f(x)$ with the x-axis is also on the graph of $g(x)$, and they have a common tangent line at this point.

$(1)$ Find the values of $a$ and $b$;

$(2)$ For any $x > 0$, compare the values of $f(x)$ and $g(x)$.

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