There are 2011 positive numbers with both their sum and the sum of their reciprocals equal to 2012.  Let $x$ be one of these numbers.  Find the maximum value of $x + \frac{1}{x}.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

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