1. The problem is to multiply the fractions $\frac{2}{3}$ and $\frac{3}{14}$.\n\n2. The formula for multiplying fractions is:\n$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$\n\n3. Applying the formula, we multiply the numerators and denominators:\n$$\frac{2}{3} \times \frac{3}{14} = \frac{2 \times 3}{3 \times 14}$$\n\n4. This gives:\n$$\frac{6}{42}$$\n\n5. Now simplify the fraction by dividing numerator and denominator by their greatest common divisor (GCD), which is 6:\n$$\frac{\cancel{6}^1}{\cancel{42}^7} = \frac{1}{7}$$\n\n6. So, the product of $\frac{2}{3}$ and $\frac{3}{14}$ is $\frac{1}{7}$.