
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

Ms. $D$ places 13 cards with numbers $1, 2, \dots, 13$ in ascending order clockwise around a circle, all initially face up. Each turn, Ms. $D$ can turn over a card to its face-down side (once a card is face down, it cannot be flipped again). Ms. $D$'s flipping rule is: if a face-up card has the number $A$, and the card two positions away from it is also face up, then Ms. $D$ can turn over the card with number $A$. For example, if the cards with numbers 9 or 2 are face up, Ms. $D$ can flip the card numbered 12 (assuming the card numbered 12 is still face up). Ms. $D$ succeeds if she turns over 12 cards, leaving only one card face up. How many different sequences of flips can Ms. $D$ perform to succeed?

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