The Many Faces Of An Inequality

Algebra Level 5

\[\sqrt{x^2+2xy+4y^2}+\sqrt{4y^2+6yz+9z^2}+\sqrt{9z^2+3xz+x^2}\]

If real numbers \(x,y,z\) satisfy \(x+2y+3z=2\sqrt3\), find the minimum value of the above expression.

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