Inspired by shubhendra Singh

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.

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