
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 $f(x)=\ln(ax+ \frac{1}{2})+ \frac{2}{2x+1}$.
$(1)$ If $a > 0$, and $f(x)$ is monotonically increasing on $(0,+\infty)$, find the range of the real number $a$;
$(2)$ Is there a real number $a$ such that the minimum value of $f(x)$ on $(0,+\infty)$ is $1$? If it exists, find the value of the real number $a$; if not, please explain why.

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