
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 function $f(x)=\log _{2}x+t(t$ is a positive constant), and $f(t)=3$.  
$(1)$ Find the analytical expression of $f(x)$;  
$(2)$ If the range of the function $g(x)=\left\{\begin{array}{l}f(x),0＜x≤1\\{x}^{2}-2ax,x＞1\end{array}\right.$ is $R$, find the range of real number $a$.

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