In a circle with radius \( r \), a chord of length \( 2 \) is drawn. Let \( P \) be the midpoint of this chord. A line is drawn through \( P \) and intersects the circle at points \( A \) and \( B \) such that \( PA \cdot PB = 1 \). Find the radius \( r \) in simplest fractional form, \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers. Compute \( m + n \). Let's think step by step and output the final answer within \boxed{}.
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