AMO-Bench 2025 Problem 23

It is known that the equation in $u, v$, $u^2 - 3v^4 = 1$, has at most four positive integer solutions; and the equation in $\lambda, \iota$, $4\lambda^2 - 3\iota^2 = 1$, has at most four positive integer solutions. If a prime $p$ satisfies $p \equiv 5 \pmod{8}$, and integers $x, y$ satisfy $$\left(\frac{1+x^2}{2} + \frac{\sqrt{ 9x^4 + \frac{4x^6}{3} + 9 + 18x^2\,}}{6}\right)^{ 1/3} + \left(\frac{1+x^2}{2} - \frac{\sqrt{ 9x^4 + \frac{4x^6}{3} + 9 + 18x^2\,}}{6}\right)^{ 1/3} = 2py^2 - x^3,$$ find the maximum value of $x+y$.