Making his way to infinity

Geometry Level 3

An insect starts moving from the origin, \((0,0)\) along the straight line in zig-zag manner. He first moves to \( A = (3,3) \), changes directions and walks half that distance to \(B\), changes direction and walks a further half of the distance to \(C\), so on and so forth.

If it ultimately ends at a point \( (a,b) \), then find the value of \(a+b\).

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