Let \(A_1,A_2,\ldots,A_n\) be subsets of \(\{1,2,\ldots,n\}\) such that \(i\notin A_i\) for all \(i\). Prove that there exist four sets \(C_1,C_2,C_3,C_4\) such that \(C_1\cup C_2\cup C_3\cup C_4=\{1,2,\ldots,n\} \) and for all \(i\) and \(j\), if \(i\in C_j\), then \( \lvert A_i\cap C_j\rvert \le \frac12 \lvert A_i\rvert\).