Determine the smallest possible real constant such that the inequality holds for all real numbers satisfying .
Determine the smallest possible real constant $C$ such that the inequality
$$|x^3 + y^3 + z^3 + 1| \leqslant C|x^5 + y^5 + z^5 + 1|$$holds for all real numbers $x, y, z$ satisfying $x + y + z = -1$.
Izvor: Srednjoeuropska matematička olimpijada 2017, ekipno natjecanje, problem 2
Školjka 
such that the inequality 
holds for all real numbers 
satisfying 
.