Let $f_n(x) = n + x^2$. Evaluate the product \( \text{gcd}\{f_{2001}(2002), f_{2001}(2003)\} \times \text{gcd}\{f_{2011}(2012), f_{2011}(2013)\} \times \text{gcd}\{f_{2021}(2022), f_{2021}(2023)\} \), where \( \text{gcd}\{x, y\} \) is the greatest common divisor of \( x \) and \( y \).

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