
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
A polynomial with rational coefficients has all of the numbers \[2+\sqrt{3}, \; 3+\sqrt{4}, \;4+\sqrt{5},\; \dots, \;1001+\sqrt{1002}\] as roots. What is the smallest possible degree of such a polynomial?

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