
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
Read the following material: Through elementary school learning, we know that fractions can be divided into "proper fractions" and "improper fractions", and improper fractions can all be converted into mixed numbers, such as: $\frac{8}{3}=\frac{{6+2}}{3}=2+\frac{2}{3}=2\frac{2}{3}$. We define: in a fraction, for a fraction with only one letter, when the power of the numerator is greater than or equal to the power of the denominator, we call it an "improper fraction"; when the power of the numerator is less than the power of the denominator, we call it a "proper fraction". For example, $\frac{{x-1}}{{x+1}},\frac{{{x^2}}}{{x-1}}$, these fractions are improper fractions; and for example: $\frac{3}{{x+1}},\frac{{2x}}{{{x^2}+1}}$, these fractions are proper fractions. Similarly, improper fractions can also be converted into mixed fractions (i.e., the form of the sum of a polynomial and a proper fraction), such as: $\frac{{x-1}}{{x+1}}=\frac{{({x+1})-2}}{{x+1}}=1-\frac{2}{{x+1}}$. Solve the following problems:  
$(1)$ The fraction $\frac{1}{{3{x^2}}}$ is ______ $(fill in "proper fraction" or "improper fraction")$;  
$(2)$ Convert the improper fraction $\frac{{4a+1}}{{2a-1}}$ into the form of the sum of a polynomial and a proper fraction: $\frac{{4a+1}}{{2a-1}}=\_\_\_\_\_\_.$ If the value of the improper fraction $\frac{{4a+1}}{{2a-1}}$ is a positive integer, then the value of the integer $a$ is ______;  
$(3)$ Convert the improper fraction $\frac{{{x^2}-2x-1}}{{x-1}}$ into a mixed fraction (write out the complete process).

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