
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 as follows: \(b_1 = b_2 = b_3 = 1\), and for all positive integers \(n\), \(b_{n+3} = b_{n+2} + 2b_{n+1} + b_n\). Given that \(b_{28} = 121393\), \(b_{29} = 196418\), and \(b_{30} = 317811\), find the remainder when \(\displaystyle \sum_{k=1}^{28}b_k\) is divided by 1000.

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