Greedy Calvinosaurus Dinosaur

Greedy Calvinosaurus Dinosaur is threatening to gobble you up, unless you can tell him what is the largest possible value of \[ \gcd (122 + n^2, 122 + (n+1)^2 ) ,\] as \(n\) ranges over all positive integers.

Quick, to save your life, what is the answer?

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