A Sidewinder Sequence

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Consider the recursive sequence defined by

\[a_1=1;\; a_2=2;\; a_n=a_{n-1}\sqrt{3}-a_{n-2}\text{ for }n\ge 3.\]

Then \(a_{2014}\) can be expressed as \(m+n\sqrt{p}\), where m, n, and p are integers, and p is positive and square-free. Find \(m+n+p\).

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