Inspired By Brian Charlesworth!

Let \(S\) be the set of all distinct Pythagorean Triplets \((a,b,c)\) where \(a,b,c\) are positive integers and \(a<b<c\) such that \(ab=10(a+b+c)\).

Let the sum of all distinct values of \(c\) of the elements of \(S\) be \(x\).

Let the number of elements or Pythagorean Triplets present in the set \(S\) be \(y\).

Let the sum of all distinct Inradii of all such Right Angled Triangles formed by the Pythagorean Triplets present in \(S\) be \(z\).

Find the value of \(x+y+z\).


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