Assumptions "Without Loss of Generality" (WLOG)

Let $S_1$, $S_2$, $\dots$, $S_{125}$ be 125 sets of 5 numbers each, comprising $625$ distinct numbers. Let $m_i$ be the median of $S_i$. Let $M$ be the median of $m_1$, $m_2$, $\dots$, $m_{125}$. What is the greatest possible number of the 625 numbers that are less than $M$?

(Hint: We may assume "without loss of generality" that $m_1 > m_2 > \dots > m_{125}$.)

Source: Math Prize for Girls