Suppose $x$ is a random real number between $1$ and $4$, and $y$ is a random real number between $1$ and $9$. If the expected value of \[ \left\lceil \log_2 x \right\rceil - \left\lfloor \log_3 y \right\rfloor \] can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, compute $100m + n$.

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