IMAS 2021–2022 Junior Division, Round 2 Question 8

Define $f(x)=\frac1{\sqrt[3]{x^2+2x+1}+\sqrt[3]{x^2-1}+\sqrt[3]{x^2-2x+1}}$, find the value of $f(1)+f(3)+\ldots+f(2k-1)+f(999)$.