
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 $a \gt 0$, the function $f\left(x\right)=\left\{\begin{array}{l}\left|{log}_{2}x\right|,x≥a\\ \frac{x-2}{x-3},x＜a且x≠3\end{array}\right.$, when $a=2$, the range of $f\left(x\right)$ is ______; if there does not exist $x_{1}$, $x_{2}(x_{1}\neq x_{2})$ such that $f(x_{1})=f(x_{2})$, then the range of real number $a$ is ______.

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