
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

Determine all non-empty subsets $A, B, C$ of $\mathbf{N}^{*}$ such that:
1. $A \cap B = B \cap C = C \cap A = \varnothing$;
2. $A \cup B \cup C = \mathbf{N}^{*}$;
3. For all $a \in A, b \in B, c \in C$, we have \(a + c \in A\), \(b + c \in B\), and \(a + b \in C\).

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