
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 $f(x)=(\log_{2}\frac{x}{8})\cdot [\log_{2}(2x)]$ and the function $g(x)=4^{x}-2^{x+1}-3$.
(1) Find the range of the function $f(x)$.
(2) If the inequality $f(x)-g(a)\leqslant 0$ always holds for any real number $a\in[\frac{1}{2},2]$, determine the range of real numbers $x$.

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