AMO-Bench 2025 Problem 13

Let $\{a_k\}$ be a sequence of non-negative integers, and $\{b_k\}$ be a sequence of positive real numbers. Given that: 1.$$\left(\sum_{n=1}^{2023} a_n b_n\right)\left( \sum_{k=0}^{673} \frac{1}{(k+675)(1348-k)} \right)= \sum_{k=1}^{674}\frac{1}{k(2k-1)}$$ 2.$$\frac{\sum_{k=1}^{1364176}\frac{1}{k(2k-1)}}{\sum_{k=0}^{1364175} \frac{1}{(k+1364177)(2728352-k)}} = 1+\sum_{n=1}^{2023} a_n^2 b_n^2$$ 3.$$\left( \sum_{k=0}^{5519455420} \frac{1}{(k+5519455422)(11038910842-k)} \right)\left[\left(\sum^{2023}_{n=1}2a^3_nb^3_n \right)-1\right] =\sum_{k=1}^{5519455421} \frac{1}{k(2k-1)}$$ Find the minimum value of $$\sum_{n=1}^{2023} a_n^4 b_n^4$$