Let \( { f(x) = \text{max}\left\{ { x }^{ 2 },{ \left( 1-x \right) }^{ 2 },2x(1-x) \right\} }\).

Determine the area of the region bounded bt the curves \(y=f(x)\), \(x\)-axis, \(x=0, x=1\).

If the area is of the form \(\frac a b\) for coprime positive integers \(a,b\), find \(a+b\).

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