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Algebra Level 5

The system of equations \[ 4a^4+36a^2b^2+9b^4=20a^2+30b^2-1\] \[2a^3b+3ab^3=5ab\] has \(n\) nonnegative (meaning both \(a\) and \(b\) are nonnegative) real solutions. Let these solutions be \((a_1,b_1),(a_2,b_2),\cdots (a_n,b_n)\). The value of the sum \[\sum_{i=1}^n a_i+b_i\] can be expressed in the form \(\dfrac{a+b\sqrt{c}}{d}\) where \(\gcd (a,b,d)=1\) and \(c\) is squarefree. Find \(a+b+c+d\).

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