A digital display shows the current date as an $8$-digit integer consisting of a $4$-digit year, followed by a $2$-digit month, followed by a $2$-digit date within the month. For example, Arbor Day this year is displayed as 20230428. Find the number of dates in $2023$ for which each digit appears an even number of times in the 8-digit display for that date.

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