AMO-Bench 2025 Problem 32

Let $x$ and $y$ be positive rational numbers that satisfy: $$1 + \sqrt{1 - 2y} = y + \frac{4x - 2 + 2\sqrt{1 - 2x}}{3 - 5x - \sqrt{1 - 2x}} + \sqrt[3]{\frac{1 + x^2}{2} + \frac{1}{6} \cdot \sqrt{9x^4 + \frac{4x^6}{3} + 9 + 18x^2}}+ \sqrt[3]{\frac{1 + x^2}{2} - \frac{1}{6} \cdot \sqrt{9x^4 + \frac{4x^6}{3} + 9 + 18x^2}}$$ Given that $kx$ and $ky$ are both positive integers, find the minimum value of the positive integer $k$.