Let \(a, b, c,\) and \(d\) be real numbers that satisfy the system of equations \begin{align*} a+b&=-3\\ ab+bc+ca&= -4\\ abc+bcd+cda+dab&=14\\ abcd&=30. \end{align*}There exist relatively prime positive integers \(m\) and \(n\) such that $$a^2 + b^2 + c^2 + d^2 = \frac{m}{n}.$$Find \(m + n\).