Given any two positive real numbers $x$ and $y$, then $x\Diamond y$ is a positive real number defined in terms of $x$ and $y$ by some fixed rule.  Suppose the operation $x\Diamond y$ satisfies the equations $(x\cdot y)\Diamond y=x(y\Diamond y)$ and $(x\Diamond 1)\Diamond x=x\Diamond 1$ for all $x,y>0$.  Given that $1\Diamond 1=1$, find $19\Diamond 98$.

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