If $x = \frac{\sqrt{5} + 1}{\sqrt{5} - 1}$ and $y = \frac{\sqrt{5} - 1}{\sqrt{5} + 1}$, find the value of $x^2 + y^2 + xy$.
Options:
- A: $8$
- B: $12$
- C: $16$
- D: $18$
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