
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 that $a \in \mathbb{R}$, $f(x) = ax^2 - \ln{x}$, and $g(x) = e^x - ax$,

1. Find the monotonic intervals of $f(x)$ when the slope of the tangent line to the curve $y = f(x)$ at the point $(1, f(1))$ is greater than $-1$.
2. If $f(x) \cdot g(x) > 0$ holds for all $x \in (0, +\infty)$, find the range of values for the real number $a$.

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