Trigo is not such easy!!

Algebra Level 5

If \(\lfloor 2 \cos x \rfloor + \lfloor \sin x \rfloor =-3\) for \(x \in [0, 2 \pi]\), then the range of the function \(f(x)= \sin x+\sqrt 3 \cos x\) can be written as \([a, b] \). Find \(a^2 + b^2\).

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

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