Shifting the numbers, part 5

$\frac{\overline{A_1A_2A_3\ldots A_{n-1}A_n}}{\overline{A_2A_3\ldots A_{n-1}A_nA_1}}=\frac{12}{17}$

What is the smallest value of $$n$$ such that there exists solutions to the above equation?

Remark: You may try a related problem here.

This question is part of the set Shifting the numbers.

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