
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
On a blackboard, there are 2000 numbers written as \(1, 2, 3, \cdots, 2000\). Each time, you are allowed to erase two numbers \(a, b\) (where \(a \geq b\)) and write \(a\), \(b\), \( \sqrt{a b} \), \( \sqrt{a b} \) – the number \(\sqrt{a b}\) is written twice. After performing this operation 8000 times, 10000 numbers are obtained. Can these 10000 numbers all be less than 500?

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