
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 $f(x)=\ln(x+1)+\ln(1-x)+a(x+1)$ where $a > 0$.

$(1)$ When $a=1$, find the intervals of monotonicity for the function $f(x)$.

$(2)$ If the maximum value of $f(x)$ on the interval $(-1,0]$ is $1$, find the value of the real number $a$.

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