Those are some special dice!

Calvin has a collection of special weighted dice that all share special properties:

  • They're all 4 sided dice
  • Any one die has distinct positive integers on each of its faces
  • No pair of dice have all their numbers exactly the same
  • The probability of rolling number \(x\) on any one of the dice is \(\frac {1}{x}\)

Let \(a\) be the maximum number of dice in the collection and let \(S\) be the sum of all the faces of all the dice in the maximum collection size.

Find \(a+S\).


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