Given $$-\frac{1}2<a<\alpha<0<\beta<b<+\frac{1}2$$ $$+\frac{1}2<c<\gamma<1<\delta<d<+\frac{3}2$$ I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f([a,b])\subseteq[\alpha,\beta]$, $f([c,d])\subseteq[\gamma,\delta]$. What is minimum degree polynomial that is needed and maximum degree that will suffice?

I believe Chebyshev polynomials play a role here. What is degree as function of $a,b,c,d,\alpha,\beta,\gamma,\delta$ that is necessary and sufficient using Chebyshev polynomials?

Is there an explicit formula (not a computational solution) that gives a good enough answer within constants?