
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
Define a sequence where $b_1 = b_2 = b_3 = 2$, and for all positive integers $n$, $b_{n+3} = b_{n+2} + b_{n+1} + b_n$. Given that $b_{25} = 2555757$, $b_{26} = 4700770$, and $b_{27} = 8651555$, find the remainder when $\displaystyle \sum_{k=1}^{25}b_k$ is divided by 1000.

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