Root Flipping, Vieta Jumping, and Low-Degree Curves

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[Romania TST $2004$ ] Find all integer values the expression $\frac{a^2+b^2+1}{ab-1}$ can assume for $ab\neq 1$ where a, b are natural numbers.

Choose all correct answers

Option A: A

0

Option B: B

1

Option C: C

2

Option D: D

3

Option E: E

4

Option F: F

5

Option G: G

6

Option H: H

7

Option I: I

8

Option J: J

9

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