Don't trip

Let \(S\) be the set of all distinct Pythagorean triples \((a,b,c)\), where \(a,b,c\) are positive integers with \(a \lt b \lt c\), such that \(ab = 6(a + b + c).\)

Find the sum of all distinct values for \(c\) of the elements of \(S.\)

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