
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
Let the function $f(x)=(1+x)^{2}-2\ln (1+x)$.  
$(1)$ Find the intervals of monotonicity for the function $f(x)$;  

$(2)$ When $x\in\left[\frac{1}{e}-1, e-1\right]$, the inequality $f(x) < m$ always holds, find the range of the real number $m$.

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