
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
Let \([x]\) denote the greatest integer less than or equal to \(x\), e.g. \([\pi] = 3\), \([5.31] = 5\), and \([2009] = 2009\). Evaluate \(\left[\sqrt{2009^2 + 1} + \sqrt{2009^2 + 2} + \cdots + \sqrt{2009^2 + 4018}\right]\).

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