Let \[{ a }_{ n }=\sum _{ k=1 }^{ n }{ (k+2)\sum _{ j=1 }^{ k }{ \sum _{ i=1 }^{ j }{ i!({ i }^{ 2 }+i+1) } } } \] and \[{ b }_{ n }={ a }_{ n+1 }-{ a }_{ n } .\] Compute \(b_4\).

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**Notation:** \(!\) is the factorial notation. For example, \(8! = 1\times2\times3\times\cdots\times8 \).

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