
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 $f(x) = \frac{e^x - a}{x}$ and $g(x) = a \ln x + a$.
1. When $a=1$, determine the intervals of monotonicity for $F(x) = f(x) - g(x)$.
2. If $y=f(x)$ is always above $y=g(x)$ for $x>1$, find the range of the real number $a$.

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