
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: In a fraction, when the degree of the numerator is greater than or equal to the degree of the denominator, we call it a "false fraction." For example, $\frac{x-1}{x+1}, \frac{x^2}{x+2}$ are false fractions; when the degree of the numerator is less than the degree of the denominator, we call it a "true fraction." For example, $\frac{1}{x+1}, -\frac{2x}{{x}^{2}-1}$ are true fractions. We know that false fractions can be converted into mixed numbers. For example, $\frac{8}{3}=\frac{3×2+2}{3}=3\frac{2}{3}$. Similarly, false fractions can also be converted into "mixed fractions," which are in the form of the sum of a polynomial and a true fraction. For example, $\frac{x^2+2x-1}{x+2}=\frac{x(x+2)-1}{x+2}=x-\frac{1}{x+2}; \frac{x^2}{x+2}=\frac{(x^2+2x)-2x}{x+2}=\frac{x(x+2)-2x-4}{x+2}=\frac{x(x+2)}{x+2}\frac{-2(x+2)+4}{x+2}=x-2+\frac{4}{x+2}$. Please answer the following questions based on the above material:  
$(1)$ Fill in the blanks: ① The fraction $\frac{2}{x+2}$ is a ______ fraction (fill in "true" or "false"); ② Convert the following false fraction into the form of the sum (difference) of a polynomial and a true fraction: $\frac{{x}^{2}+2x-13}{x-3}=\_\_\_\_\_\_+\_\_\_\_\_\_.$  
$(2)$ Convert the fraction $\frac{{x}^{2}+2x-13}{x-3}$ into the form of the sum (difference) of a polynomial and a true fraction, and find the integer values of $x$ for which this fraction is an integer.  
$(3)$ A three-digit number $m$, the units digit is twice the hundreds digit, and another two-digit number $n$, the tens digit is the same as the hundreds digit of $m$, and the units digit is the same as the tens digit of $m$. If the square of this three-digit number can be divided by this two-digit number, find the two-digit number $n$ that satisfies the condition.

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