Just the right number to divide

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 \(\displaystyle \sum_{k=1}^{28} a_{k}\) is divided by 1000.

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