1. **State the problem:** Simplify the expression $$6 \cdot \left(\frac{1}{6} + \frac{3}{7}\right) + \frac{2}{7}$$. 2. **Use the distributive property:** Multiply 6 by each term inside the parentheses: $$6 \cdot \frac{1}{6} + 6 \cdot \frac{3}{7} + \frac{2}{7}$$ 3. **Simplify each multiplication:** $$\cancel{6} \cdot \frac{1}{\cancel{6}} + \frac{18}{7} + \frac{2}{7} = 1 + \frac{18}{7} + \frac{2}{7}$$ 4. **Add the fractions with common denominator 7:** $$\frac{18}{7} + \frac{2}{7} = \frac{18 + 2}{7} = \frac{20}{7}$$ 5. **Add the whole number and the fraction:** $$1 + \frac{20}{7} = \frac{7}{7} + \frac{20}{7} = \frac{27}{7}$$ 6. **Final answer:** $$\boxed{\frac{27}{7}}$$