
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
A sequence where all terms are either 0 or 1 is called a 0,1 sequence. Let \( A \) be a finite 0,1 sequence, and let \( f(A) \) denote the 0,1 sequence obtained by replacing each 1 in \( A \) with 0,1 and each 0 in \( A \) with 1,0. For example, \( f((1,0,0,1)) = (0,1,1,0,1,0,0,1) \). Let \( f^n(A) \) denote \( f(f(\cdots f(A) \cdots)) \) (applying \( f \) \( n \) times).

Question: How many consecutive pairs of 0s are there in \( f^n((1)) \)?

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