\[\large f(x) = \frac{1}{ 2+ \lfloor \sin x\rfloor}\]

If \( a_1 \) is the largest value of \( f(x) \) above and \(a_{n+1} = \dfrac{ (-1)^{n+2} }{n+1} + a_{n} \), for \(n \ge 1\), find the value of \(\displaystyle \lim_{n \to \infty} a_{n}\).

Notation: \( \lfloor \cdot \rfloor \) denotes the floor function.

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