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

Let's think step by step and output the final answer within \boxed{}.
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