GAIC Math 2024 Problem 86

Given $n$ is positive integer, for $i=1,2, \cdots, n$, positive integer $a_{i}$ and positive even number $b_{i}$ satisfy $0<\frac{a_{i}}{b_{i}}<1$, and for any positive integers $i_{1}, i_{2}\left(1 \leq i_{1}<i_{2} \leq n\right), a_{i_{1}} \neq a_{i_{2}}$ and $b_{i_{1}} \neq b_{i_{2}}$ at least one holds true. If for any positive integer $n$ and all positive integers $a_{i}$ and positive even numbers $b_{i}$ satisfy the above conditions, we all have $\frac{\sum_{i=1}^{n} b_{i}}{n^{\frac{3}{2}}} \geq c$, then the maximum value of real number $c$ is _____.