Compute the sum of all positive integers $n < 200$ such that $\text{gcd}(n, k) \neq 1$ for every $k \in \{2 \cdot 11 \cdot 19, 3 \cdot 13 \cdot 17, 5 \cdot 11 \cdot 13, 7 \cdot 17 \cdot 19\}$.

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