A binary operation $♢$ has the properties that $a♢(b♢c)=(a♢b)\cdot c$ and that $a♢a=1$ for all nonzero real numbers $a,b,c$. (Here $\cdot$ represents multiplication). The solution to the equation $2024♢(8♢x)=50$ can be written as $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. What is $p+q$?