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 \(\sum^{28}_{k=1} a_k\) is divided by 1000.

(第二十四届AIME2 2006 第11题)