A list of integers has a mode of $32$, mean $22$ and median $m$. $m$ is one of the numbers on the list.
The smallest number on the list is $10$.
If $m$ were replaced with $m + 10$, the mean of the new list would be $24$.
If $m$ were instead replaced with $m-8$, the median of the new list would be $m - 4$.
What is $m$?
This problem is adapted from the World Mathematics Championships