
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 $f(x) = (\log_2{x} - 2)(\log_4{x} - \frac{1}{2})$:
(1) When $x \in [1, 4]$, find the range of the function.
(2) If $f(x) > m \cdot \log_4{x}$ holds for all $x \in [4, 16]$, determine the range of $m$.

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