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.