
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 $f(x) = (a - \frac{1}{2})x^2 + \ln{x}$ where $a \in \mathbb{R}$.

(1) When $a = 1$, find the maximum and minimum values of $f(x)$ on the interval $[1, e]$.

(2) Find the range of $a$ such that the graph of the function $f(x)$ is always below the line $y = 2ax$ on the interval $(1, +\infty)$.

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