1. Stating the problem: We need to multiply the mixed numbers $3 \frac{1}{5}, 6 \frac{2}{3},$ and $12$, and then divide $\frac{1}{6}$ by $\frac{12}{13}$ and further divide by $\frac{13}{9}$.\n\n2. Convert mixed numbers to improper fractions:\n$3 \frac{1}{5} = \frac{16}{5}$ because $3 \times 5 + 1 = 16$.\n$6 \frac{2}{3} = \frac{20}{3}$ because $6 \times 3 + 2 = 20$.\n3. Multiply the fractions:\n$$\frac{16}{5} \times \frac{20}{3} \times 12 = \frac{16 \times 20 \times 12}{5 \times 3 \times 1}$$\nSimplify denominator: $5 \times 3 = 15$.\nSimplify numerator and denominator:\n$$\frac{16 \times 20 \times 12}{15} = \frac{3840}{15}$$\nDivide numerator by denominator:\n$$3840 \div 15 = 256$$\nSo the multiplication result is 256.\n\n4. Now divide $\frac{1}{6}$ by $\frac{12}{13}$ and then by $\frac{13}{9}$:\nWhen dividing fractions, multiply by the reciprocal:\n$$\frac{1}{6} \div \frac{12}{13} = \frac{1}{6} \times \frac{13}{12} = \frac{13}{72}$$\nThen divide by $\frac{13}{9}$:\n$$\frac{13}{72} \div \frac{13}{9} = \frac{13}{72} \times \frac{9}{13} = \frac{9}{72} = \frac{1}{8}$$\n\n5. Final answers:\n- Multiplication: $256$\n- Division: $\frac{1}{8}$