These numbers... where have I seen them?

Algebra Level 5

Given that \(0<x<2,\quad 0<y<2\),

Let the minimum value of \(\sqrt { 2{ x }^{ 2 }+2{ y }^{ 2 } } +\sqrt { { y }^{ 2 }+{ x }^{ 2 }-4y+4 } +\sqrt { { x }^{ 2 }+{ y }^{ 2 }-4y-4x+8 } \) be \(a\sqrt { b } \) such that \(b\) is square free. Find \(a+b\).

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