Let $a,b,c,d$ positive real numbers, such that $$\frac1a+\frac1b+\frac1c+\frac1d=4.$$ Prove inequality $$\sqrt[3]{\frac{a^3+b^3}{2}}+\sqrt[3]{\frac{b^3+c^3}{2}}+\sqrt[3]{\frac{c^3+d^3}{2}}+\sqrt[3]{\frac{d^3+a^3}{2}} \le 2(a+b+c+d)-4$$
My work so far:
$$1=\frac{4}{\frac1a+\frac1b+\frac1c+\frac1d}\le \sqrt[4]{abcd}\le\frac{a+b+c+d}{4}$$