If $y - x,$ $y - 2x,$ and $y - kx$ are all factors of
\[x^3 - 3x^2 y + p_1 xy^2 + p_2 y^3,\]then find $k$.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

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