
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 inequality $\left(\log_{2}x\right)^{2}-\log_{2}x^{2}\leqslant 0$, the solution set is $A$, and the function $f(x)=4^{x- \frac{1}{2}}-a\cdot2^{x}+ \frac{a^{2}}{2}+1, x\in A(a\in R)$

$(1)$ Find the domain of $f(x)$;

$(2)$ Find the minimum value $g(a)$ of the function $f(x)$.

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