A set of six distinct positive integers is required to satisfy all the following conditions:
• The range of the set is 40.
• The second-largest value is exactly 12 less than the largest value.
• The arithmetic mean of the smallest and the largest values is 30.
(i) Let the largest value be L and the smallest value be S. Using the given information, express S in terms of L and hence determine the numerical values of L and S.
(ii) Construct one possible data set of six distinct positive integers that meets all the stated conditions.
(iii) Verify explicitly that the data set you have constructed in part (ii) has the required range of 40.