
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 function $f(x) = 3 - 2\log_{2}x$, $g(x) = \log_{2}x$.

(1) If $x \in [1, 8]$, find the range of the function $h(x) = (f(x) + 1)g(x)$;

(2) Find the maximum value of the function $M(x) = \begin{cases} g(x), & f(x) \geq g(x) \\ f(x), & f(x) < g(x) \end{cases}$;

(3) If the inequality $f(x^2)f(\sqrt{x}) \geq kg(x)$ holds for any $x \in [1, 8]$, find the range of the real number $k$.

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