Given any two positive real numbers $x$ and $y$, the operation $x \Diamond y$ is defined by some fixed rule. It satisfies the following equations for all $x, y > 0$:

1. $(x \cdot y) \Diamond y = x(y \Diamond y)$
2. $(x \Diamond 1) \Diamond x = x \Diamond 1$

Additionally, given that $1 \Diamond 1 = 1$, find the value of $19 \Diamond 98$.

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