
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 $f(x) = e^{x} - \frac {x}{4}$, where $e$ is the base of natural logarithms,
(1) Let $g(x) = xf'(x)$ (where $f'(x)$ is the derivative of $f(x)$), determine the monotonicity of $g(x)$ on $(0, +\infty)$.
(2) If $F(x) = \ln x - af(x) + 1$ has no zeroes, determine the range of $a$.

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