Algebra Level 3

If a periodic non-zero function $$f(x)$$ exists such that $$\sqrt{3}f(x) = f(x - 1) + f(x + 1),$$ what is its fundamental period?

Note: A function $$g(x)$$ is periodic if there exists some $$T$$ such that $$g(x) = g(x + T)$$ for all $$x.$$ The fundamental period is the minimum positive $$T$$ such that this property holds.

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