A binary operation $\diamondsuit$ has the properties that $a\diamondsuit (b\diamondsuit c) = (a\diamondsuit b)\cdot c$ and that $a\diamondsuit a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\cdot$ represents multiplication). The solution to the equation $2016 \diamondsuit (6\diamondsuit x)=100$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is the value of $p+q$?

Let's think step by step and output the final answer within \boxed{}.
user