Real numbers $r$ and $s$ are roots of $p(x)=x^3+ax+b$, and $r+4$ and $s-3$ are roots of $q(x)=x^3+ax+b+240$. Find the sum of all possible values of $|b|$.
I assumed that the other root of p(x) would be t. From here, I used Vieta's formulas to get $-rst=b$ and $rs+st+rt=a$. Then, I assumed that the third root of q(x) would be y. Then, I used vieta's formulas to get that $-(r+4)(s-3)y=b+240$ and that $(r+4)(s-3)+y(s-3)+y(r+4)=a$. I am not sure how to proceed from here.