The positive integers $x_1, x_2, \ldots, x_7$ satisfy $x_6 = 144$ and $x_{n+3} = x_{n+2}(x_{n+1} + x_n)$ for $n = 1, 2, 3, 4$. Find $x_7$.

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