
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 permutation of the numbers \(1, 2, 3, \ldots, n\) is called amusing if every number, starting from the second one on the left, is either greater than all the numbers to its left or less than all the numbers to its left. For example, the permutation 3, 2, 1, 4, 5, 6 is amusing, while the permutation 3, 1, 2, 4, 5, 6 is not. Find the number of all distinct amusing permutations of the numbers \(1, 2, 3, \ldots, n\).

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