
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 $f(x)=x^{2}+a\ln x$ $(a\in\mathbb{R})$,

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

$(2)$ If the function $f(x)$ is increasing on the interval $(1,+\infty)$, find the range of $a$;

$(3)$ Discuss the monotonicity of the function $g(x)=f(x)-(a+2)x$.

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