
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
Find all arithmetic sequences \( a_{1}, a_{2}, \cdots, a_{n} \) such that each term is an integer, and
$$
\begin{array}{l}
a_{1}^{3} + a_{2}^{3} + a_{3}^{3} = a_{4}^{3}, \\
a_{1} + a_{2} + \cdots + a_{n} = 500.
\end{array}
$$

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