
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 $f(x)=\ln (x+1)+ax$.  
$(1)$ When $x=0$, the function $f(x)$ attains its maximum value, find the value of the real number $a$;  
$(2)$ If there exists $x\in[1,2]$ such that the inequality $f′(x)\geqslant 2x$ holds, where $f′(x)$ is the derivative of $f(x)$, find the range of values for the real number $a$;  
$(3)$ Determine the intervals of monotonicity for the function $f(x)$.

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