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$S$ is a set given by $S=\{1,2,3,\dots,4n\}$, where $n$ is a natural number. $S$ is partitioned into $n$ disjoint subsets $A_{1},A_{2},A_{3}\dots,A_{n}$ each containing four elements. It is given that in everyone of these subsets there is one element, which is the arithmetic mean of the other three elements of the subsets. Which of the following statements is then true?

- $n\neq1$ and $n\neq2$
- $n\neq1$ but can be equal to $2$
- $n\neq2$ but can be equal to $1$
- It is possible to satisfy the requirement for $n=1$ as well as for $n=2$