AMO-Bench 2025 Problem 31

Let $(a_i, b_i)$, for $1 \leq i \leq 2025$, be 2025 distinct points on the curve $$3x^2 + 3y^2 = 8 + 4xy + 2(x + y)|x - y|$$ Let $d_i$ be the distance from the point $(a_i, b_i)$ to the line $x + y = 6$, and let $\left\lfloor d_i \right\rfloor$ denote the greatest integer not exceeding $d_i$.

It is given that: $$r_i = \left\lfloor \frac{21 - 15\sqrt{2}}{2} \left( d_i^3 - 3d_i^2 - 6d_i^2 \cdot \sqrt{2} + 18d_i + 6d_i \cdot \sqrt{2} + \frac{28}{3} + \frac{16\sqrt{2}}{3} \right) \right\rfloor$$ and that: $$\sum_{i=1}^{2025} r_i^2 = 2043, \sum_{i=1}^{2025} r_i = 2021$$ Find the maximum value of $$\sum_{i=1}^{2025} r_i^4$$