
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 a function \( f(x) = x^2 - a^2 \ln x - ax \) where \( a \in \mathbb{R} \).
1. If \( f(x) \geq 0 \) always holds, find the range of \( a \).
2. Let \( g(x) = f(x) + ax \). If \( g(x) \) has two zeros in the interval \([ \frac{1}{e}, e ] \), find the range of \( a \).

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