
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 $f(x) = x - \frac{a}{x} - (a+1)\ln x$ ($a \in \mathbb{R}$).
(I) When $a = \frac{1}{2}$, find the intervals of monotonicity for the function $f(x)$.
(II) Determine if there exists a real number $a$ such that $f(x) \leq x$ always holds; if such an $a$ exists, find the range of values for $a$; if not, explain why.

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