
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 $f(x)=\ln(x+1)-ax+2$.  
$(1)$ If $a=2$, find the equation of the tangent line to $f(x)$ at $x=0$.  
$(2)$ For $x\geq 0$, if $f(x)+2x+x\ln(x+1)\geq 0$ always holds, find the maximum integer value of $a$.

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