Limit it all!

Calculus Level 3

\(f(n)=\lim _{ n\rightarrow \infty }{ \{ } { (\sqrt { 2 } +1) }^{ n }\} .{ (-1) }^{ [{ (\sqrt { 2 } +1) }^{ n }] }\)
Here \(n\) is an integer and \([.]\) denotes 'greatest integer function' and {.} denotes 'fractional part function'.
Find the value of \(|f(200)+f(311)+f(434)+f(507)|-f(3)\).

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