
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: \( a_1=a_2=a_3=a_4=1 \), and for all positive integers \( n \), \( a_{n+4}=a_{n+3}+a_{n+2}+a_{n+1}+a_n \). Given that \( a_{26}= 354224848179261915075 \), \( a_{27}= 573147844013817084101 \), and \( a_{28}= 927372692193078999176 \), find the remainder when \( \sum_{k=1}^{28}a_k \) is divided by 1000.

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