
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
Let  $n$  be a positive integer. Initially, a bishop is placed in each square of the top row of a  $2^n \times 2^n$ chessboard; those bishops are numbered from  $1$  to  $2^n$  from left to right. A *jump* is a simultaneous move made by all bishops such that each bishop moves diagonally, in a straight line, some number of squares, and at the end of the jump, the bishops all stand in different squares of the same row.

Find the total number of permutations  $\sigma$  of the numbers  $1, 2, \ldots, 2^n$  with the following property: There exists a sequence of jumps such that all bishops end up on the bottom row arranged in the order  $\sigma(1), \sigma(2), \ldots, \sigma(2^n)$ , from left to right.

*Israel*

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