CGMO 2018 Problem 6

Given $k \in \mathbb{N}^+$ . A sequence of subset of the integer set $\mathbb{Z} \supseteq I_1 \supseteq I_2 \supseteq \cdots \supseteq I_k$ is called a $k-chain$ if for each $1 \le i \le k$ we have (i) $168 \in I_i$ ; (ii) $\forall x, y \in I_i$ , we have $x-y \in I_i$ . Determine the number of $k-chain$ in total.