
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
In a new sequence, the first term is $b_1 = 2010$ and the second term is $b_2 = 2012$. Furthermore, the sequence is defined such that
\[b_n + b_{n + 1} + b_{n + 2} = 2n + 3 \text{ for all } n \ge 1.\]
Determine $b_{1000}$.

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