For each ordered pair of real numbers $(x,y)$ satisfying\[\log_2(2x+y) = \log_4(x^2+xy+7y^2)\]there is a real number $K$ such that\[\log_3(3x+y) = \log_9(3x^2+4xy+Ky^2).\]Find the product of all possible values of $K$.
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