A pair of positive integers $(m,n)$ is called *compatible* if $m \ge \frac{1}{2} n + 7$ and $n \ge \frac{1}{2} m + 7$. A positive integer $k \ge 1$ is called *lonely* if $(k,\ell)$ is not compatible for any integer $\ell \ge 1$. Find the sum of all lonely integers.

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