
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
A new definition is given: rounding a non-negative number to the nearest integer is denoted as $\left[a\right]$, that is, when $n$ is a non-negative integer, if $n-\frac{1}{2}≤a＜n+\frac{1}{2}$, then $\left[a\right]=n$. For example: $\left[0\right]=\left[0.499\right]=0$, $\left[0.54\right]=\left[1.499\right]=1$, $\left[2\right]=2$, $\left[3.51\right]=\left[4.32\right]=4$. Solve the following problems:<br/>$(1)$ Fill in the blank: If $\left[a\right]=2$, then the range of real number $a$ is ______;<br/>$(2)$ Given the system of inequalities about $x$: $\left\{\begin{array}{l}{\frac{1}{2}x-1≥1-\frac{3}{2}x}\\{[b-1]-x＞0}\end{array}\right.$ has exactly $2$ integer solutions, find the range of $b$.<br/>$(3)$ Find all non-negative real numbers $c$ that satisfy $[c]=\frac{5}{4}c$.

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