
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 $f(x) = \ln x - ax$ ($a \in \mathbb{R}$), $g(x) = -x^2 + 2x + 1$,
(I) Find the intervals of monotonicity for $f(x)$;
(II) If for any $x_1 \in [1, e]$, there always exists $x_2 \in [0, 3]$ such that $f(x_1) = g(x_2)$, find the range of the real number $a$.

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