
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
Given a permutation \(a_{1}, a_{2}, \ldots, a_{21}\) of the numbers \(1, 2, \ldots, 21\), such that \(\left|a_{20}-a_{21}\right| \geq \left|a_{19}-a_{21}\right| \geq \cdots \geq \left|a_{1}-a_{21}\right|\), find the number of such permutations.

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