1. **State the problem:** Calculate the product of the fractions $\frac{6}{7} \times \frac{14}{21} \times \frac{14}{3}$.\n\n2. **Recall the multiplication rule for fractions:** When multiplying fractions, multiply the numerators together and the denominators together: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}.$$\n\n3. **Apply the rule:** Multiply all numerators and denominators:\n$$\frac{6}{7} \times \frac{14}{21} \times \frac{14}{3} = \frac{6 \times 14 \times 14}{7 \times 21 \times 3}.$$\n\n4. **Calculate numerator and denominator:**\nNumerator: $6 \times 14 = 84$, then $84 \times 14 = 1176$.\nDenominator: $7 \times 21 = 147$, then $147 \times 3 = 441$.\nSo the fraction is $$\frac{1176}{441}.$$\n\n5. **Simplify the fraction:** Find the greatest common divisor (GCD) of 1176 and 441.\n- Prime factors of 1176: $2^3 \times 3 \times 7^2$.\n- Prime factors of 441: $3^2 \times 7^2$.\nCommon factors: $3 \times 7^2 = 3 \times 49 = 147$.\n\n6. **Divide numerator and denominator by 147:**\n$$\frac{1176 \div 147}{441 \div 147} = \frac{8}{3}.$$\n\n**Final answer:** $$\frac{6}{7} \times \frac{14}{21} \times \frac{14}{3} = \frac{8}{3}.$$