
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

Notice that \((n+1)^{2}-(n-1)^{2}=4n\), i.e., the numbers \(a=4n\) multiple of four are represented as the difference of the squares of two integers, and thus, with the condition of zero, are represented as the sum or difference of four squares. If \(a=4n \pm 1\), then

\[ a=(n+1)^{2}-(n-1)^{2} \pm 1^{2} \]

If \(a=4n+2\), then it can be represented as the sum or difference of four squares:

\[ a=(n+1)^{2}-(n-1)^{2}+1^{2}+1^{2} \]

The number 2021 can be written using the representation \(2021=4 \cdot 505+1\), therefore, for example, \(2021=506^{2}-504^{2}+1^{2}\).

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