
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 $f(x)=\frac{1+\ln x}{x}$.  
(1) If the function $f(x)$ has an extremum in the interval $(a,a+\frac{1}{2})$, find the range of values for the positive real number $a$;  
(2) If for $x \geqslant 1$, the inequality $f(x) \geqslant \frac{k}{x+1}$ always holds, find the range of values for the real number $k$.

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