
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 that $2^{x}\leqslant 256$ and $\log \_{2}x\geqslant \dfrac {1}{2}$,
$(1)$ Find the range of $x$;
$(2)$ Find the maximum and minimum values of the function $f(x)=\log \_{2}( \dfrac {x}{2}) \cdot \log \_{\sqrt {2}}( \dfrac {\sqrt {x}}{2})$.

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