
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 $a_1,a_2,\dots,a_{2023}$ be a strictly increasing sequence of positive integers such that $a_1+a_2+\cdots+a_{2023}=2023^{2023}$. Determine the remainder when $a_1^3 - a_2^3 + a_3^3 - a_4^3 + \cdots - a_{2022}^3 + a_{2023}^3$ is divided by $6$.

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