There are real numbers \(a, b, c,\) and \(d\) such that \(-20\) is a root of \(x^3 + ax + b\) and \(-21\) is a root of \(x^3 + cx^2 + d.\) These two polynomials share a complex root \(m + \sqrt{n} \cdot i,\) where \(m\) and \(n\) are positive integers and \(i = \sqrt{-1}.\) Find \(m+n.\)