
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 $f(x) = x + \frac{a}{x}$ and $g(x) = x^2 - 2ax + 2$.

(1) Let proposition $p$ be: $\forall x \in [2, +\infty), f(x) \geqslant 2$ always holds; let proposition $q$ be: the function $g(x)$ is monotonically increasing on $[2, +\infty)$. If both $p$ and $q$ are true, find the range of the real number $a$.

(2) Let $F(x) = \begin{cases} f(x), & x \geqslant 2 \\ g(x), & x < 2 \end{cases}$. If for $\forall x_1 \in [2, +\infty)$, there always exists $x_2 \in (-\infty, 2)$ such that $F(x_1) = F(x_2)$ holds, find the range of the real number $a$.

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