A geometric sequence $\left\{ a_{n} \right\}$ satisfies
$$\sum_{n=1}^{\infty} \left(\left| a_{n} \right| + a_{n}\right) = \frac{40}{3}, \quad \sum_{n=1}^{\infty} \left(\left| a_{n} \right| - a_{n}\right) = \frac{20}{3}$$
The inequality
$$\lim_{n \rightarrow \infty} \sum_{k=1}^{2n} \left((-1)^{\frac{k(k+1)}{2}} \times a_{m+k}\right) > \frac{1}{700}$$
is satisfied. What is the sum of all natural numbers $m$ satisfying this inequality? [4 points]