
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 non-negative real number $x$, the value obtained by "rounding" $x$ to the nearest integer is denoted as $ \lt x \gt $. That is: when $n$ is a non-negative integer, if $n-\frac{1}{2}≤x＜n+\frac{1}{2}$, then $ \lt x \gt =n$; conversely, when $n$ is a non-negative integer, if $ \lt x \gt =n$, then $n-\frac{1}{2}≤x＜n+\frac{1}{2}$. For example: $ \lt 0 \gt = \lt 0.49 \gt =0$, $ \lt 0.67 \gt = \lt 1.46 \gt =1$, $ \lt 3.5 \gt = \lt 4.13 \gt =4$, $ \lt 2 \gt =2$. Solve the following problems:
  
$(1)$ Fill in the blanks: ① $ \lt \pi -1 \gt =\_\_\_\_\_\_ (\pi $ is the constant pi); ② $＜\sqrt{15}＞=$______;
  
$(2)$ If $ \lt x-1 \gt =4$, then the range of real number $x$ is ______;
  
$(3)$ If the system of inequalities about $x$ $\left\{\begin{array}{l}{\frac{2x-7}{3}≤x-2}\\{＜a＞-x＞0}\end{array}\right.$ has exactly $3$ integer solutions, then the range of $a$ is ______;
  
$(4)$ Find all non-negative real numbers $x$ that satisfy $＜x＞=\frac{3}{5}x+1$, and write down the necessary steps.

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