HMMT 2023 U35

The Fibonacci numbers are defined recursively by $F_{0}=0, F_{1}=1$, and $F_{i}=F_{i-1}+F_{i-2}$ for $i \geq 2$. Given 30 wooden blocks of weights $\sqrt[3]{F_{2}}, \sqrt[3]{F_{3}}, \ldots, \sqrt[3]{F_{31}}$, estimate the number of ways to paint each block either red or blue such that the total weight of the red blocks and the total weight of the blue blocks differ by at most 1 .

Submit a positive integer $E$. If the correct answer is $A$, you will receive $\left\lfloor 25 \min \left((E / A)^{8},(A / E)^{8}\right)\right\rfloor$ points. (If you do not submit a positive integer, you will receive zero points for this question.)