
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
Consider the sequence of $n$-tuples $R_{0}, R_{1}, \cdots$ defined as follows. Each element of $R_{i}$ belongs to the set $\{A, B, C\}$. If $R_{i} = \left(x_{1}, x_{2}, \cdots, x_{n}\right)$, then $R_{i+1} = \left(y_{1}, y_{2}, \cdots, y_{n}\right\}$. 

If $x_{i} = x_{(i + 1 \mod n)}$, then $y_{i} = x_{i}$. Otherwise, $y_{i} \in \{A, B, C\} \setminus \{x_{i}, x_{(i + 1 \mod n)}\}$ for $i = 1, 2, \cdots, n$.

For example, if $R_{0} = (A, A, B, C)$, then $R_{1} = (A, C, A, B)$.

1. Find all positive integers $n$ for which there exists $m \in \mathbf{N}^{*}$ such that for any $R_{0}$, we have $R_{m} = R_{0}$.
2. For $n = 3^{k}$, $k \in \mathbf{N}^{*}$, find the smallest $m \in \mathbf{N}^{*}$ such that for any $R_{0}$, we have $R_{m} = R_{0}$.

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