
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: $a_1 = a_2 = a_3 = 1$, and for all positive integers $n$, the relationship $a_{n+3} = a_{n+2} + 2a_{n+1} + a_n$ holds. Given that $a_{28}= 6090307$, $a_{29}=11201821$, and $a_{30}=20603361$, find the remainder when $\displaystyle \sum_{k=1}^{29}a_k$ is divided by 1000.

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