
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 $u(x) = x\ln x - \ln x$, $v(x) = x - a$, and $w(x) = \frac{a}{x}$, all with the domain $A = \{x \mid x > 1\}$.

(1) If $u(x) \geq v(x)$ holds true for all $x$ in the domain, and the set of all such real numbers $a$ forms the set $B$, determine the relationship between sets $A$ and $B$, and explain the reasoning.
(2) Let $G(x) = [u(x) - w(x)][v(x) - \frac{w(x)}{2}]$, is there an $m \in \mathbb{N}^*$ such that for all real numbers $a \in (m, +\infty)$, the function $G(x)$ has exactly two zeroes? If yes, find the smallest such positive integer $m$; if no, explain the reasoning. (Reference data: $e \approx 2.7183$, $\ln(\sqrt{2} + 1) \approx 0.8814$)

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