
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
A sequence is defined by $b_1=b_2=1$, $b_3=2$, and for all positive integers $n$, $b_{n+3}=b_{n+2}+b_{n+1}+b_n$. Given that $b_{28}= 12202338$, $b_{29}=22404141$, and $b_{30}=41207902$, find the remainder when $\displaystyle \sum_{k=1}^{28}b_k$ is divided by 1000.

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