Let $X = \{-5,-4,-3,-2,-1,0,1,2,3,4,5\}$ and $S = \{(a,b) \in X \times X : x^2 + ax + b \text{ and } x^3 + bx + a \text{ have at least a common real zero.}\}$.

How many elements are there in $S$?

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