Consider the polynomial \( P(x) = ax^3 + bx^2 + cx + d \) where \( a, b, c, \) and \( d \) are integer coefficients. If \( P(1) = 10 \) and \( P(-1) = -6 \), find all possible sets of coefficients \( (a, b, c, d) \) that satisfy these conditions. How many such sets exist? Let's think step by step and output the final answer within \boxed{}.
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