
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 a positive integer \( k \). If the sequence of subsets \( I_{1} \supseteq I_{2} \supseteq \cdots \supseteq I_{k} \) of the integer set \( \mathbf{Z} \) satisfies that for \( i=1,2, \cdots, k, \) \( 168 \in I_{i} \), and for any \( x, y \in I_{i} \), we have \( x-y \in I_{i} \) (where \( x \) and \( y \) can be equal), then the sequence \( \left(I_{1}, I_{2}, \cdots, I_{k}\right) \) is called a \( k \)-chain on \( \mathbf{Z} \). How many \( k \)-chains are there on \( \mathbf{Z} \)? Please explain your reasoning.

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