
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 \( x_{i} \in \{-1,1\}, i=1,2,\cdots,2021 \), and \( x_{1}+x_{2}+\cdots+x_{k} \geq 0 \) for \( k=1,2,\cdots,2020 \), with \( x_{1}+x_{2}+\cdots+x_{2021}=-1 \). How many ordered arrays \( (x_{1}, x_{2}, \cdots, x_{2021}) \) are there?

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