
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  $n$ ,  $a_{n+3}=a_{n+2}+a_{n+1}+a_n$ . Given that  $a_{28}=6090307$ ,  $a_{29}=11201821$ , and  $a_{30}=20603361$ , find the remainder when  $\displaystyle \sum^{28}_{k=1} a_k$  is divided by 1000.

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