These numbers... where have I seen them?

Algebra Level 5

Given that 0<x<2,0<y<20<x<2,\quad 0<y<2,

Let the minimum value of 2x2+2y2+y2+x24y+4+x2+y24y4x+8\sqrt { 2{ x }^{ 2 }+2{ y }^{ 2 } } +\sqrt { { y }^{ 2 }+{ x }^{ 2 }-4y+4 } +\sqrt { { x }^{ 2 }+{ y }^{ 2 }-4y-4x+8 } be aba\sqrt { b } such that bb is square free. Find a+ba+b.

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