AIME 2021b Problem 7

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 $m + n$.