Monocarp invented a new operation on integers: $$$n \rightarrow x$$$, which he calls a cyclic shift. The operation is defined only for positive integers $$$n$$$ that do not contain zeros in their decimal representation, and for positive integers $$$x$$$. It is defined as follows: let the decimal representation of $$$n$$$ be $$$\overline{n_1 n_2 \dots n_{m-1} n_m}$$$, then:

$$$$$$n \rightarrow x = \begin{cases} \overline{n_m n_1 n_2 \dots n_{m - 1}}, & \text{if } x = 1 \\ (n \rightarrow (x - 1)) \rightarrow 1, & \text{otherwise} \end{cases} $$$$$$

In other words, $$$n \rightarrow x$$$ is the integer you get if you perform the following operation $$$x$$$ times: move the rightmost digit in the decimal representation of $$$n$$$ to the leftmost position, shifting all other digits one position to the right.

For example, $$$123 \rightarrow 1 = 312$$$, $$$123 \rightarrow 2 = 231$$$, $$$123 \rightarrow 9 = 123$$$, $$$1 \rightarrow 1 = 1$$$.

Monocarp is interested in how similar this operation is to addition. Given an integer $$$n$$$, he wants to find all values of $$$x$$$ such that $$$n \rightarrow x = n + x$$$.