
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 \( a_{1}, a_{2}, \cdots, a_{21} \) be a permutation of \( 1, 2, \cdots, 21 \) that satisfies
\[ \left|a_{20} - a_{21}\right| \geq \left|a_{19} - a_{21}\right| \geq \left|a_{18} - a_{21}\right| \geq \cdots \geq \left|a_{1} - a_{21}\right|. \]
Determine the number of such permutations.

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