$\frac{1}{a_3+a_4}+\frac{1}{a_1+a_2}\ge \frac{4}{a_1+a_2+a_3+a_4} $ Prove that $$\frac{1}{a_3+a_4}+\frac{1}{a_1+a_2}\ge \frac{4}{a_1+a_2+a_3+a_4} $$
I am supposed to use AM-HM but I don't how I should. By AM-HM
We have $$\frac{1}{a_3+a_4}+\frac{1}{a_1+a_2}\ge \frac{2}{a_1+a_2+a_3+a_4}  $$
Did I do something wrong?

Hint : Let $a_1+a_2=x$ and $a_3+a_4=y$. It remains to prove,
$$\frac{1}{x}+\frac{1}{y}\ge \frac{4}{x+y}\;\; \Leftrightarrow\;\; (x+y)\left(\frac{1}{x}+\frac{1}{y}\right)\ge 4$$
The A.M-H.M inequality for two terms,
$$\frac{x+y}{2}\ge \dfrac{1}{\dfrac{\frac{1}{x}+\frac{1}{y}}{2}}$$
