
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 the smallest positive integer $n$ such that for any $n$ different numbers $a_1, a_2, \dots, a_n$ in the interval $(1, 1000)$, there are always two numbers $a_i, a_j$ (where $1 \leq i, j \leq n$) that satisfy the inequality
\[ 0 < a_i - a_j < 1 + 3 \sqrt[3]{a_i a_j} . \]

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