Circles \(\omega_1\), \(\omega_2\), and \(\omega_3\) each have radius \(4\) and are placed in the plane so that each circle is externally tangent to the other two. Points \(P_1\), \(P_2\), and \(P_3\) lie on \(\omega_1\), \(\omega_2\), and \(\omega_3\) respectively such that \(P_1P_2=P_2P_3=P_3P_1\) and line \(P_iP_{i+1}\) is tangent to \(\omega_i\) for each \(i=1,2,3\), where \(P_4 = P_1\). See the figure below. The area of \(\triangle P_1P_2P_3\) can be written in the form \(\sqrt{a}+\sqrt{b}\) for positive integers \(a\) and \(b\). What is \(a+b\)?

\(\textbf{(A) }546\qquad\textbf{(B) }548\qquad\textbf{(C) }550\qquad\textbf{(D) }552\qquad\textbf{(E) }554\)

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