
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 for any \( n \in \mathbb{Z} \),

$$
(n+1)^3 + (n-1)^3 + (-n)^3 + (-n)^3 = 6n.
$$

That is, any integer of the form \( a = 6n \) can be represented as the sum of the cubes of four integers, and therefore, considering zero, of five integers. Numbers of the form \( a = 6n \pm 1 \) can be represented as

$$
a = (n+1)^3 + (n-1)^3 + (-n)^3 + (-n)^3 + (\pm 1)^3.
$$

Numbers of the form \( a = 6n + 2 = 6(n-1) + 8 \) can be represented as the sum of the cubes of five integers:

$$
a = n^3 + (n-2)^3 + (-n+1)^3 + (-n+1)^3 + 2^3.
$$

For numbers of the form \( a = 6n - 2 = 6(n+1) - 8 \) the representation is:

$$
a = (n+2)^3 + n^3 + (-n-1)^3 + (-n-1)^3 + (-2)^3.
$$

Finally, for \( a = 6n + 3 = 6(n-4) + 27 \) the representation is:

$$
a = (n-3)^3 + (n-5)^3 + (-n+4)^3 + (-n+4)^3 + 3^3.
$$

The representation of the number \( a = 2020 = 337 \cdot 6 - 2 \) can be obtained using the formula for \( n = 337 \):

$$
2020 = (339)^3 + 337^3 + (-338)^3 + (-338)^3 + (-2)^3.
$$

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