
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
Let  $k$  be a positive integer. A sequence of integers  $a_1, a_2, \cdots$  is called  $k$ -pop if the following holds: for every  $n \in \mathbb{N}$ ,  $a_n$  is equal to the number of distinct elements in the set  $\{a_1, \cdots , a_{n+k} \}$ . Determine, as a function of  $k$ , how many  $k$ -pop sequences there are.

*Proposed by Sutanay Bhattacharya*

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