
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
How many integer lattice points (points of the form \((m, n)\) for integers \(m\) and \(n\)) lie inside or on the boundary of the disk of radius 2009 centered at the origin?

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