
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 $f(x)=\frac{1}{x}+a\ln x$, where $a\in R$.
(1) Find the interval(s) where the function $f(x)$ is monotonically decreasing.
(2) When $x\in [\frac{1}{2}, 1]$, the minimum value of $f(x)$ is $0$. Find the value of the real number $a$.

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