Let \(A_1, \ldots, A_n\) ( \(n \geq 3\) ) be finite sets of natural numbers. Prove that

\[

\frac{1}{n} \sum_{i=1}^n\left|A_i\right|+\frac{1}{\left(\begin{array}{l}

n \\

3

\end{array}\right)} \sum_{i<j<k}\left|A_i \cap A_j \cap A_k\right| \geq \frac{2}{\left(\begin{array}{l}

n \\

2

\end{array}\right)} \sum_{i<j}\left|A_i \cap A_j\right| .

\]