
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 $f(x)=\ln x+\ln (2-x)+ax (a > 0)$.

(1) Find the domain of the function $f(x)$.

(2) When $a=1$, find the monotonic intervals of $f(x)$.

(3) If the maximum value of $f(x)$ on $(0,1]$ is $\frac{1}{2}$, find the value of $a$.

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