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