NL-NWO 2019 Problem 1

A complete number is a $9$ digit number that contains each of the digits $1$ to $9$ exactly once. The difference number of a number $N$ is the number you get by taking the differences of consecutive digits in $N$ and then stringing these digits together. For instance, the difference number of $25143$ is equal to $3431$ . The complete number $124356879$ has the additional property that its difference number, $12121212$ , consists of digits alternating between $1$ and $2$ . Determine all $a$ with $3 \le a \le 9$ for which there exists a complete number $N$ with the additional property that the digits of its difference number alternate between $1$ and $a$ .