
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
Find the greatest common divisor (GCD) of the numerator and the denominator. Using the properties of the GCD:

$$
\begin{aligned}
& \text {GCD}(5n-7, 3n+2) = \text {GCD}(5n-7-(3n+2), 3n+2) = \\
& = \text{GCD}(2n-9, 3n+2) = \text{GCD}(2n-9, n+11) = \\
& \quad = \text{GCD}(n-20, n+11) = \text{GCD}(31, n+11).
\end{aligned}
$$

Since 31 is a prime number, we get that \(\text{GCD}(31, n+11) = 1\) or \(\text{GCD}(31, n+11) = 31\). The first case does not apply because it does not reduce the fraction. The second case is possible only if \(n+11\) is divisible by 31, i.e., for \(n = 31k - 11\), where \(k \in \mathbb{Z}\). In this case, the fraction is simplified only by 31. There cannot be any other common divisors between the numerator and the denominator since 31 is a prime number.

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