
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: \( b_1 = b_2 = b_3 = 2 \), and for all positive integers \( n \), \( b_{n+3} = b_{n+2} + b_{n+1} + b_n \). Given that \( b_{30} = 24791411 \), \( b_{31} = 45765219 \), and \( b_{32} = 84215045 \), find the remainder when \( \sum_{k=1}^{30} b_k \) is divided by 1000.

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