
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 $f(x)=\frac{ax}{1+x^{2}}+1 (a\neq 0)$.
(1) When $a=1$, find the equation of the tangent line to the graph of $f(x)$ at the point $(0,1)$;
(2) Find the intervals where the function $f(x)$ is monotonically increasing or decreasing;
(3) If $a>0$, $g(x)=x^{2}e^{mx}$, and for any $x_{1}$, $x_{2}\in[0,2]$, $f(x_{1})\geq g(x_{2})$ always holds, find the range of the real number $m$.

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