
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 $f(x)=\ln x$, and $g(x)= \frac{1}{3}x^{3}+ \frac{1}{2}x^{2}+mx+n$, the line $l$ is tangent to the graphs of functions $f(x)$ and $g(x)$ at the point $(1,0)$.
$(1)$ Find the equation of line $l$ and the expression for $g(x)$;
$(2)$ If $h(x)=f(x)-g'(x)$, where $g'(x)$ is the derivative of $g(x)$, find the maximum value of the function $h(x)$.

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