For integers a, b, c, and d, let f(x)=x^{2}+a x+b and g(x)=x^{2}+c x+d. Find the number of ordered triples (a, b, c) of integers with absolute values not exceeding 10 for which there is an integer d such that g(f(2))=g(f(4))=0.
For integers a, b, c, and d, let f(x)=x^{2}+a x+b and g(x)=x^{2}+c x+d. Find the number of ordered triples (a, b, c) of integers with absolute values not exceeding 10 for which there is an integer d such that g(f(2))=g(f(4))=0.