
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 $f(x)=x^{3}+ax^{2}-a^{2}x+2$.

1. If $a=1$, find the equation of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$.
2. If $a\neq 0$, find the intervals where the function $f(x)$ is monotonic.
3. If the inequality $2x\ln x\leqslant f'(x)+a^{2}+1$ always holds, find the range of real numbers for $a$.

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