\[\dfrac{1}{2}+\dfrac{1}{4}-\dfrac{3}{8}-\dfrac{7}{16}+\dfrac{5}{32}+\dfrac{33}{64}+\dfrac{13}{128}+\cdots+\dfrac{g(n)}{2^n}+\cdots\]

The numerators \(g(n)\) are given by \(g(n)=g(n-1)-4g(n-2)\) for \(n>2\), with \(g(1)=g(2)=1\).

If \(M\) and \(m\) are the supremum and the infimum, respectively, of the partial sums of the series above, find \(M+m\). As your answer, enter \(\lfloor 1000(M+m) \rfloor\).

Enter 2016 if you come to the conclusion that the supremum or the infimum fail to exist.

\[ \]

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

This problem is a follow-up to this question.

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