\[\sqrt{a^2+ab+2b^2}+\sqrt{b^2+bc+2c^2}+\sqrt{c^2+ac+2a^2}\]

Given that \(a,b\) and \(c\) are real numbers satisfying \(a+b+c\geq1007 \), find the minimum value of the expression above.

**Bonus**: Try to solve this problem using vectors.

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