
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 $f(x)=x^{2}-x+b$, and $f(\log_{2}a)=b$, $\log_{2}[f(a)]=2$ ($a\neq 1$).
$(1)$ Find the minimum value of $f(\log_{2}x)$ and the corresponding value of $x$;
$(2)$ For what value of $x$ does $f(\log_{2}x) > f(1)$ and $\log_{2}[f(x)] < f(1)$?

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