
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 interval $P=[\frac{1}{2},2]$, the domain of the function $y=\log_{2}(ax^{2}-2x+2)$ is $Q$.

$(1)$ If $P\cap Q\neq \varnothing$, find the range of the real number $a$;

$(2)$ If the equation $\log_{2}(ax^{2}-2x+2)=2$ has a solution in the interval $[\frac{1}{2},2]$, find the range of the real number $a$.

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