Numbers In a circle

Let $A_n$ denote the numbers that can be on either side of the number $n$ .

$A_1 = \{3,5,7\}$

$A_2 = \{6,9\}$

$A_3 = \{1,5,8\}$

$A_4 = \{7,9\}$

$A_5 = \{3,6,8\}$

$A_6 = \{2,5,7\}$

$A_7 = \{1,4,6,9\}$

$A_8 = \{3,5,9\}$

$A_9 = \{2,4,7,8\}$

Thus the answer must contain $629$ or $926$ as they are the only possible neighbours of $2$ .

The should also contain $947$ or $749$ for the same reason.

Since $9$ is common the answer must contain $62947$ or $74926$ .

Now $7$ cannot have $6$ or $9$ as neighbours it must have $1$ as the other neighbour.

Thus the answer must contain $629471$ or $174926$ .

Similarly $6$ cannot have $7$ as a neighbor thus it should have $5$ as it's other neighbour.

Thus the answer must contain $5629471$ or $1749265$ .

Now $8$ must have $3$ and $5$ as neighbours.

Thus the answer must contain $385629471$ or $174926583$ .

Thus rearranging the sequences to start with $1$ we get $138562947$ and $174926583$ .

These are the possible answers to the question. You should modify the question to make the current answer the only possible answer like beginning from $1$ and smallest amomg all possible arrangement.