
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 $f(x)=\ln x$, $g(x)=f(x)+ax^{2}+bx$, where the tangent line to the graph of the function $g(x)$ at the point $(1,g(1))$ is parallel to the $x$-axis.

$(1)$ Determine the relationship between $a$ and $b$;

$(2)$ If $a\geqslant 0$, discuss the monotonicity of the function $g(x)$.

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