
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
In the interval $[1,1000]$, select $\mathrm{n}$ different numbers $\mathrm{a}_{1}, \mathrm{a}_{2}, \ldots, \mathrm{a}_{n}$. There always exist two numbers $\mathrm{a}_{i}, \mathrm{a}_{j}$ such that $0 < \left|a_{i} - a_{j}\right| < 1 + 3 \sqrt[3]{a_{i} a_{j}}$. Find the smallest possible value of $n$.

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