
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 \(\left(x_{1}, x_{2}, \cdots, x_{20}\right)\) be a permutation of \((1, 2, \cdots, 20)\), such that it satisfies
\[ 
\sum_{i=1}^{20}\left(\left|x_{i}-i\right|+\left|x_{i}+i\right|\right) = 620.
\]
How many such permutations exist?

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