
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 functions $f(x)=({\log_2}\frac{x}{8})\cdot[{\log_2}(2x)]$ and $g(x)=4^{x}-2^{x+1}-3$. Find:

$(1)$ The range of function $f(x)$;

$(2)$ If the inequality $f(x)-g(a)\leqslant 0$ holds for any real number $a$ in the interval $[\frac{1}{2},2]$, find the range of real numbers for $x$.

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