
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 $f(x)=\frac{1+\ln x}{2ax}$ ($a\neq 0$), where $a$ is a constant.

1. Find the monotonic intervals of $f(x)$;
2. Let $a=\frac{1}{2}$. If there exist $x\_1$, $x\_2$ in the interval $(1,+\infty)$, and $x\_1\neq x\_2$ such that the inequality $|f(x\_1)-f(x\_2)|\geqslant k|\ln x\_1-\ln x\_2|$ holds, find the range of $k$.

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