Consider the polynomial \( P(x) = x^3 + ax^2 + bx + c \), where \( a, b, c \) are integers. Suppose \( P(x) \) has integer roots and the sum of these roots is equal to the 10th harmonic number \( H_{10} \). Find the value of \( a \). Let's think step by step and output the final answer within \boxed{}.
user