1. **State the problem:** Calculate the value of $$1 \frac{3}{7} \times \frac{1}{4} \times \left( \frac{2}{3} + \frac{2}{5} \right)$$. 2. **Convert mixed number to improper fraction:** $$1 \frac{3}{7} = \frac{10}{7}$$ because $1 \times 7 + 3 = 10$. 3. **Add the fractions inside the parentheses:** $$\frac{2}{3} + \frac{2}{5} = \frac{2 \times 5}{3 \times 5} + \frac{2 \times 3}{5 \times 3} = \frac{10}{15} + \frac{6}{15} = \frac{16}{15}$$. 4. **Rewrite the expression with improper fractions:** $$\frac{10}{7} \times \frac{1}{4} \times \frac{16}{15}$$. 5. **Multiply the numerators and denominators:** $$\frac{10 \times 1 \times 16}{7 \times 4 \times 15} = \frac{160}{420}$$. 6. **Simplify the fraction by dividing numerator and denominator by their greatest common divisor (20):** $$\frac{\cancel{160}^{8}}{\cancel{420}^{21}} = \frac{8}{21}$$. 7. **Final answer:** $$\boxed{\frac{8}{21}}$$