
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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. 

Present the answer in LaTex format: \boxed{Your answer}
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