
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 $b_1=2$, $b_2=3$, $b_3=5$, and for all $n \geq 1$, $b_{n+3} = b_{n+2} + b_{n+1} + b_n$. Given $b_{18}= 7083$, $b_{19}=13013$, and $b_{20}=23919$, find the remainder when $\sum_{k=1}^{19} b_k$ is divided by 1000.

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