RU-ALLRUSSIAN 2021 Problem 104

Given a natural number $n>4$ and $2n+4$ cards numbered with $1, 2, \dots, 2n+4$ . On the card with number $m$ a real number $a_m$ is written such that $\lfloor a_{m}\rfloor=m$ . Prove that it's possible to choose $4$ cards in such a way that the sum of the numbers on the first two cards differs from the sum of the numbers on the two remaining cards by less than $$\frac{1}{n-\sqrt{\frac{n}{2}}}$$ .