
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
The sequence of numbers \( a_{1}, a_{2}, \ldots, a_{2022} \) is such that \( a_{n} - a_{k} \geq n^{3} - k^{3} \) for any \( n \) and \( k \) where \( 1 \leq n \leq 2022 \) and \( 1 \leq k \leq 2022 \). Furthermore, \( a_{1011} = 0 \). What values can \( a_{2022} \) take?

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