
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 functions $f(x) = ax - \ln x$ and $g(x) = \frac{\ln x}{x}$, where $a \in \mathbb{R}$:
1. Discuss the monotonicity intervals of $f(x)$ when $a = g'(1)$.
2. Determine if there exists a real number $a$ such that the minimum value of $f(x)$ is 3 for $x \in [0, e]$. If such an $a$ exists, find its value; if not, explain why.

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