
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: $a_{1}=1599$, and for $k^{2}<n \leqslant(k+1)^{2}$, $a_{n}=\left[\frac{k+1}{2}\right]+\left[\frac{k}{2}\right]+1$ where $k \in \mathbf{N}^{*}$, is there a positive integer $n$ such that the equation $\sum_{i=1}^{n} a_{i}=2006$ holds? If yes, find all such numbers $n$; if no, explain why.

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