Least Negative Most Positive?

Algebra Level 5

Let \(\alpha\), \(\beta\) and \(\gamma\) be the real roots of the cubic equation \(x^3-7x^2+ax+50=0\), and their absolute values be the roots of another cubic \(x^3-bx^2+cx-d=0\). Find the minimum integral value of \(b+d\).