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