You are given the following list of the seven numbers: \(10\), \(2\), \(5\), \(2\), \(4\), \(6\), \(2\). You are asked to choose a positive number as the eighth number so that
the mean, median, and mode of the eight numbers are distinct, and
the mean, median, and mode differ by the same amount when put in order from least to greatest.
Determine all possible choices for the eighth number.