
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 $f(x) = ax^2 - 2x + 2 + \ln x \ (a > 0)$,

1. Find the set of values for the real number $a$ such that $f(x)$ is a monotonically increasing function in its domain.
2. When $a = \frac{3}{8}$, the function $y = f(x)$ has zero points in the domain $[e^n, +\infty] \ (n \in \mathbb{Z})$. Find the maximum value of $n$.

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