
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
Let $n$ be a given positive integer. Find all integer sequences $\left(a_{1}, a_{2}, \cdots, a_{n}\right)$ satisfying the conditions:
1. $a_{1}+a_{2}+\cdots+a_{n} \geqslant n^{2}$;
2. $a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2} \leqslant n^{3}+1$.

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