
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
In each cell of a square grid of size \( \left(2^{n}-1\right) \times \left(2^{n}-1\right) \) one of the numbers 1 or -1 is placed. The arrangement is called successful if each number is equal to the product of all its neighboring numbers (neighboring numbers are considered those that are in cells with a common side). Find the number of successful arrangements.

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