RMO 2023 K3

Let $f(x)$ be a polynomial with real coefficients of degree 2. Suppose that for some pairwise distinct real numbers , $a,b,c$ we have:

$$f(a)=bc , f(b)=ac, f(c)=ab$$ Dertermine $f(a+b+c)$ in terms of $a,b,c$ .