
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
In a fraction, when both the numerator and denominator are polynomials, if the degree of the numerator is less than the degree of the denominator, the fraction is called a proper fraction. For example, the fractions $\frac{4}{x+2}$ and $\frac{3x^{2}}{x^{3}+4x}$ are proper fractions. If the degree of the numerator is not less than the degree of the denominator, the fraction is called an improper fraction. For example, the fractions $\frac{x-4}{x+2}$ and $\frac{x^{2}}{x-1}$ are improper fractions. An improper fraction can be expressed as the sum of a polynomial and a proper fraction. For example, $\frac{x-4}{x+2}= \frac{(x+2)-6}{x+2}=1- \frac{6}{x+2}$.

1. Express the improper fraction $\frac{x+1}{x-1}$ as the sum of a polynomial and a proper fraction.
2. Express the improper fraction $\frac{2x-1}{x+1}$ as the sum of a polynomial and a proper fraction.
3. If the value of the fraction $\frac{x^{2}}{x+1}$ is an integer, find the integer value of $x$.

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