We say that a four-digit number $\overline{abcd}$ is \textit{slippery} if the number $a^4 + b^3 + c^2 + d$ is equal to the two-digit number $\overline{cd}$. For example, $2023$ is slippery, since $2^4 + 0^3 + 2^2 + 3 = 23$. How many slippery numbers are there?

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