Consider the polynomial \( P(x) = x^4 + ax^3 + bx^2 + cx + d \) where \( a, b, c, \) and \( d \) are real numbers. Suppose that \( P(x) \) has roots that form an arithmetic sequence. Given that the sum of the roots is 10 and the product of the roots is 24, determine the value of \( c \). Let's think step by step and output the final answer within \boxed{}.
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