Digits in a circle

Suppose we place \(K\) distinct digits around a circle such that each pair of adjacent digits (read either clockwise or counterclockwise) forms a number that's divisible by \(7.\) What's the maximum possible value of \(K?\)

For example, the diagram to the right illustrates a solution for \(K=3.\) Note that 35, 56, and 63 are all multiples of 7.

Note: Not all pairs of digits need to be read in the same direction. Some pairs may be read clockwise and others may be read counterclockwise.

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