
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
Define a number to be an anti-palindrome if, when written in base 3 as \(a_{n} a_{n-1} \ldots a_{0}\), then \(a_{i} + a_{n-i} = 2\) for any \(0 \leq i \leq n\). Find the number of anti-palindromes less than \(3^{12}\) such that no two consecutive digits in base 3 are equal.

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