Suppose we have \(4\) circles \(P, Q, R, S\) each of radius \(1\) such that \(Q\) is centered at \((0,1)\), \(R\) at \((2,1)\), \(S\) at \((4,1)\) and with \(P\) lying in the first quadrant tangent to both \(R\) and \(S\).

Form a triangle \(\Delta ABC\) that circumscribes the \(4\) circles such that \(AB\) is tangent to \(P\) and \(Q\), \(AC\) is tangent to \(P\) and \(S\), and \(BC\) is tangent to \(Q, R\) and \(S\).

The perimeter of \(\Delta ABC\) can be written as \(a + b\sqrt{c}\), where \(a,b,c\) are positive integers with \(c\) being square-free. Find \(a + b + c\).

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