Consider the set \( A = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \). Let \( f \) and \( g \) be functions from \( A \) to \( A \) such that \( f(x) = x + 1 \) mod 9 and \( g(x) = x + 2 \) mod 9. Calculate the number of elements in the set \( \{x \in A : (f \circ g)(x) = (g \circ f)(x)\} \). Let's think step by step and output the final answer within \boxed{}.
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