FKMO 2017 Problem 2

For a positive integer $n$ , $(a_0, a_1, \cdots , a_n)$ is a $n+1$ -tuple with integer entries. For all $k=0, 1, \cdots , n$ , we denote $b_k$ as the number of $k$ s in $(a_0, a_1, \cdots ,a_n)$ . For all $k = 0,1, \cdots , n$ , we denote $c_k$ as the number of $k$ s in $(b_0, b_1, \cdots ,b_n)$ . Find all $(a_0, a_1, \cdots ,a_n)$ which satisfies $a_0 = c_0$ , $a_1=c_1$ , $\cdots$ , $a_n=c_n$ .