Four. (50 points) Take the subset $A_{i}=\left\{a_{i}, a_{i+1}, \cdots, a_{i+59}\right\}(i=1,2, \cdots$, 70 ) of the set $S=\left\{a_{1}, a_{2}, \cdots, a_{70}\right\}$, where $a_{70+i}=a_{i}$. If there exist $k$ sets among $A_{1}, A_{2}, \cdots, A_{70}$ such that the intersection of any seven of them is non-empty, find the maximum value of $k$.

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The problem is asking to find the maximum number of sets $k$ from the collection $\{A_1, A_2, \ldots, A_{70}\}$ such that the intersection of any seven of these sets is non-empty. Each set $A_i$ is defined as a subset of $S$ containing 60 consecutive elements, with the indices wrapping around modulo 70.