
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 $f(x)=\ln(x+1)-x$.  
$(1)$ Find the monotonic interval of $f(x)$;  
$(2)$ When $x>0$, if $\frac{kx^2}{{e^x}-1}-x \leq f(x)$ always holds, find the maximum value of $k$.

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