Billy wrote a sequence of five numbers on the board, each an integer between $0$ and $4$, inclusive. Penny then wrote a sequence of five numbers that measured some statistics about Billy’s sequence. In particular, Penny first wrote down the number of $0$s in Billy’s sequence. Then Penny wrote the number of $1$s in Billy’s sequence, and then the number of $2$s, the number of $3$s, and finally the number of $4$s. It turned out that Penny’s sequence was exactly the same as Billy’s! What was this sequence? Express your answer as an ordered $5$-tuple.