
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 a sequence $\{a_n\}$ whose terms are either 1 or 3. The first term is 1, and between the $k$-th 1 and the $(k+1)$-th 1, there are $2k-1$ 3s, i.e., the sequence is 1, 3, 1, 3, 3, 3, 1, 3, 3, 3, 3, 3, 1, …, etc. Find the sum of the first 2010 terms of this sequence.

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