
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 \_\_\_\_\_.

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