1. **State the problem:** Find the sum of the expressions $\left(\frac{3}{7}m - 3 + 4n\right) + \left(\frac{2}{7}m - 2n + 6\right)$. 2. **Write the expression:** $$\left(\frac{3}{7}m - 3 + 4n\right) + \left(\frac{2}{7}m - 2n + 6\right)$$ 3. **Group like terms:** $$\left(\frac{3}{7}m + \frac{2}{7}m\right) + (4n - 2n) + (-3 + 6)$$ 4. **Add the coefficients of like terms:** $$\frac{3}{7}m + \frac{2}{7}m = \frac{3+2}{7}m = \frac{5}{7}m$$ $$4n - 2n = 2n$$ $$-3 + 6 = 3$$ 5. **Write the simplified sum:** $$\frac{5}{7}m + 2n + 3$$ **Final answer:** $$\boxed{\frac{5}{7}m + 2n + 3}$$