
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
A sequence of $n$ elements $\left(a_{1}, a_{2}, \cdots, a_{n}\right)$ is called "intermittently periodic" if there exists a non-negative integer $i$ and a positive integer $p$ such that $i+2p \leq n$, and for all $j=1,2, \cdots, p$, the following condition holds:
$$
a_{i+j} = a_{i+p+j}.
$$

Let $k$ be a given positive integer. Find the smallest positive integer $n$ such that the sequence $\left(a_{1}, a_{2}, \cdots, a_{n}\right)$ with $a_{i} \in \{1,2, \cdots, k\}$ is not intermittently periodic, but its extension $\left(a_{1}, a_{2}, \cdots, a_{n+1}\right)$ is intermittently periodic for any $a_{n+1} \in \{1,2, \cdots, k\}$.

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