Two circles of radius 1 are to be constructed as follows. The center of circle \(A\) is chosen uniformly and at random from the line segment joining \((0,0)\) and \((2,0)\). The center of circle \(B\) is chosen uniformly and at random, and independently of the first choice, from the line segment joining \((0,1)\) to \((2,1)\). Find the probability that circles \(A\) and \(B\) intersect. The original answer is in the form \(\frac{k \sqrt{m} - n}{p}\), please give the value of k + m + n + p.

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