
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 $f(x) = \ln x + \frac{1}{2}ax^2 - (a+1)x$ ($a \in \mathbb{R}$).
(I) When $a=1$, find the number of zeros of the function $y=f(x)$.
(II) When $a>0$, if the minimum value of the function $y=f(x)$ on the interval $[1, e]$ is $-2$, find the value of $a$.

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