
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 that $x$ is a real number satisfying $9^x - 4 \times 3^{x+1} + 27 \leq 0$ and $f(x) = \left(\log_{2} \frac{x}{2}\right)\left(\log_{\sqrt{2}} \frac{\sqrt{x}}{2}\right)$,
(Ⅰ) Determine the range of the real number $x$;
(Ⅱ) Find the maximum and minimum values of $f(x)$, and the corresponding values of $x$.

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