
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 the sequence $\left\{a_{n}\right\}$ defined as follows:
- \(a_{1}=1599\)
- For \(k^{2} < n < (k+1)^{2}\), \(a_{n} = \left\lfloor \frac{k+1}{2} \right\rfloor + \left\lfloor \frac{k}{2} \right\rfloor + 1\) where \(k \in \mathbf{N}^{*}\),

determine whether there exists a positive integer \(n\) such that the equation \(\sum_{i=1}^{n} a_{i}=2006\) is satisfied. If such an \(n\) exists, find all possible values of \(n\); otherwise, provide an explanation.

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