
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
Given \( S_{n} = 1 + \frac{1}{1 + \frac{1}{3}} + \frac{1}{1 + \frac{1}{3} + \frac{1}{6}} + \cdots + \frac{1}{1 + \frac{1}{3} + \frac{1}{6} + \cdots + \frac{1}{k_{n}}} \), where \( k_{n} = \frac{n(n+1)}{2} \) (\( n \in \mathbf{N}_{+} \)), let \( T_{0} \) be the largest integer satisfying the inequality \( S_{2006} > T \). Among the numbers 2006, 2008, 1006, and 1005, which one is closest to \( T_{0} \)?

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