A bug (of negligible size) starts at the origin on the coordinate plane. First, it moves one unit right to \((1,0)\). Then it makes a \(90^\circ\) counterclockwise turn and travels \(\frac{1}{2}\) a unit to \(\left(1, \frac{1}{2} \right)\). If it continues in this fashion, each time making a \(90^\circ\) degree turn counterclockwise and traveling half as far as the previous move, to which point \((x, y)\) will it come closest? Provide the answer in the form of two fractions \(\left(\frac{a}{b}, \frac{c}{d}\right)\), and calculate the value of a + b + c + d.

Let's think step by step and output the final answer within \boxed{}.
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