
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
Given the Cantor set \(C\) consisting of non-negative integers whose ternary representation contains no 1s, and the similarity transformation defined for a set \(A \subseteq \mathbf{Z}\) by \(aA+b\), show the following for the unique primitive representation of \(C\):
1. \(a_i > 1\) for all \(i\);
2. All \(a_i\) are powers of 3;
3. \(a_i > b_i\) for all \(i\);
4. The unique primitive representation of \(C\) is \(C = (3C) \cup (3C + 2)\).

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