
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 $f(x)= \frac {x}{1+x}-a\ln (1+x) (a\in\mathbb{R})$, $g(x)=x^{2}e^{mx} (m\in\mathbb{R})$.  
$(1)$ When $a=1$, find the maximum value of the function $f(x)$.  
$(2)$ When $a < 0$, and for any real numbers $x_{1}$, $x_{2}\in[0,2]$, $f(x_{1})+1\geqslant g(x_{2})$ always holds, find the range of the real number $m$.

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