
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
The infinite sequence $\{a_n\}$ consists of $k$ different numbers, and $S_n$ is the sum of the first $n$ terms of $\{a_n\}$. If for any $n \in \mathbb{N}^*$, $S_n \in \{1, 3\}$, then the maximum value of $k$ is $\boxed{4}$.

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