
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 $f(x)=\ln x+ \frac{a}{x}$.

(1) When $a > 0$, find the intervals of monotonicity for $f(x)$;

(2) If the minimum value of $f(x)$ on the interval $[1,e]$ is $1$, find the value of the real number $a$; (where $e$ is the base of the natural logarithm);

(3) If $f(x) < \frac{1}{2}x$ holds true on the interval $(1,\infty)$, find the range of values for the real number $a$.

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