
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
Arrange $1,2, \cdots, k$ in a sequence such that each number is either strictly greater than all the numbers before it or strictly less than all the numbers before it. Let the number of different arrangements be $a_{k}(k=1,2, \cdots)$. What is $a_{n}$?

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