
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 $f(x)=ax^{2}+21nx$.
(1) Find the monotonic intervals of $f(x)$.
(2) If the maximum value of $f(x)$ on $(0,1]$ is $-2$, find the value of $a$.
(3) Let $g(x)=f(x)+(a-1)\ln x+1$. When $a\leqslant -2$, if for any $x_{1}$, $x_{2}\in(0,+\infty)$, there always exists $|g(x_{1})-g(x_{2})|\geqslant k|x_{1}-x_{2}|$, try to find the maximum value of $k$.

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