
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 $f(x)=\ln x + \frac{a}{x}$ (where $a \in \mathbb{R}$):
(I) If the tangent line to the function $f(x)$ at $x=1$ is parallel to the line $2x-y=0$, find the value of the real number $a$.
(II) Discuss the monotonicity of $f(x)$ on the interval $(1, +\infty)$.
(III) If there exists an $x_0 \in (1, +\infty)$ such that $f(x_0) \leq a$ holds, find the range of values for $a$.

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