Another Pythagorean Triple Problem

Let \(b\) and \(c\) be non-zero integers satisfying \({ 37 }^{ 2 }+{ b }^{ 2 }={ c }^{ 2 }\). Denote \(Z\) as the number of different triplets \((37,b,c)\) that satisfy these constraints; and denote \(C\) be the sum of all possible values for \(c\). Find \(Z+C\).

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