
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
Teacher \(D\) arranges 13 cards, numbered \(1, 2, \ldots, 13\), in increasing order in a clockwise direction around a circle. Initially, all cards are face up. During each turn, a card can be flipped face down if the card has a neighbor two positions away that is also face up. Once a card is flipped face down, it cannot be flipped again. Teacher \(D\) will be successful if she flips 12 cards face down. How many different sequences of flipping cards will result in success for Teacher \(D\)?

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