
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
Reading comprehension  
Materials: To study the relationship between the fraction $\frac{1}{x}$ and the denominator $x$, Xiao Ming made a table and obtained the following data:  

\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
x & \ldots & -4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 & \ldots \\
\hline
\frac{1}{x} & \ldots & -0.25 & -0.\overline{3} & -0.5 & -1 & \text{undefined} & 1 & 0.5 & 0.\overline{3} & 0.25 & \ldots \\
\hline
\end{array}
\]

From the table data, it can be observed that when $x > 0$, as $x$ increases, the value of $\frac{1}{x}$ decreases and approaches $0$ infinitely; when $x < 0$, as $x$ increases, the value of $\frac{1}{x}$ also decreases.  
Materials 2: For a fraction where both the numerator and denominator are polynomials, when the degree of the denominator is higher than the degree of the numerator, we call this fraction a proper fraction. When the degree of the denominator is not less than the degree of the numerator, we call this fraction an improper fraction. Sometimes, it is necessary to decompose an improper fraction into the sum of a polynomial and a proper fraction, a transformation like this is called partial fraction decomposition. For example: $\frac{{2x+1}}{{x-4}}=\frac{{2x-8+8+1}}{{x-4}}=\frac{{2x-8}}{{x-4}}+\frac{{8+1}}{{x-4}}=2+\frac{9}{{x-4}}$.  
Based on the above materials, complete the following questions:  
$(1)$ When $x > 0$, as $x$ increases, the value of $1+\frac{1}{x}$ ______ (increases or decreases);  
When $x < 0$, as $x$ increases, the value of $\frac{{x+2}}{x}$ ______ (increases or decreases);  
$(2)$ When $x > 1$, as $x$ increases, the value of $\frac{{2x+2}}{{x-1}}$ approaches a number infinitely, find this number;  
$(3)$ When $0\leqslant x\leqslant 2$, find the range of the algebraic expression $\frac{{5x-2}}{{x-3}}$.

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