KöMaL 2018 C1499

Find the largest positive integer $n$ for which there exists an appropriate order of the numbers $1,2,\ldots,n$ such that the large number obtained by writing all the numbers together in a row has the following property: for any pair $a$, $b$ of successive digits, at least one of the two-digit numbers $\overline{ab}$, $\overline{ba}$ is a prime.