
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
For a function $f(x)$ defined on $D$, if there exists a real number $x_{0}$ such that $f(x_{0})=x_{0}$, then $x_{0}$ is called a fixed point of the function $f(x)$. Given the function $f(x)={\log_2}[{a\cdot{4^{x-\frac{1}{2}}}-(a-1){2^{x-1}}+\frac{a}{2}+\frac{1}{4}}]$.<br/>$(1)$ If $a=0$, find the fixed points of $f(x)$;<br/>$(2)$ If the function $f(x)$ has exactly two fixed points $x_{1}$ and $x_{2}$, and $0 \lt x_{1} \lt x_{2}$, find the range of positive number $a$.

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