
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
Let $f(x)=(\log_{2}x-2)(\log_{4}x-\frac{1}{2})$.  
 $(1)$ When $x\in[1,4]$, find the range of this function;  
 $(2)$ If $f(x) > m\log_{4}x$ holds for $x\in[4,16]$, find the range of $m$.

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