For every positive integer k, let \mathbf{T}_{k}=(k(k+1), 0), and define \mathcal{H}_{k} as the homothety centered at \mathbf{T}_{k} with ratio \frac{1}{2} if k is odd and \frac{2}{3} is k is even. Suppose P=(x, y) is a point such that
\left(\mathcal{H}_{4} \circ \mathcal{H}_{3} \circ \mathcal{H}_{2} \circ \mathcal{H}_{1}\right)(P)=(20,20)
What is x+y?