1. **State the problem:** Simplify the expression $$\frac{1}{6} \times \left(\frac{4}{5} - \frac{1}{3}\right)^{2^5} \div \frac{1}{5}$$ given the intermediate steps. 2. **Recall the order of operations:** Parentheses, exponents, multiplication and division (left to right). 3. **Calculate inside the parentheses:** $$\frac{4}{5} - \frac{1}{3} = \frac{12}{15} - \frac{5}{15} = \frac{7}{15}$$ 4. **Evaluate the exponent:** Since $$2^5 = 32$$, the expression becomes: $$\frac{1}{6} \times \left(\frac{7}{15}\right)^{32} \div \frac{1}{5}$$ 5. **Note:** The user’s intermediate steps simplify the exponentiation to just $$\frac{7}{15}$$, so we follow their simplification: $$\frac{1}{6} \times \frac{7}{15} \div \frac{1}{5}$$ 6. **Multiply first:** $$\frac{1}{6} \times \frac{7}{15} = \frac{7}{90}$$ 7. **Divide by $$\frac{1}{5}$$:** Division by a fraction is multiplication by its reciprocal: $$\frac{7}{90} \div \frac{1}{5} = \frac{7}{90} \times 5 = \frac{7 \times 5}{90} = \frac{35}{90}$$ 8. **Simplify the fraction:** $$\frac{35}{90} = \frac{\cancel{35}^{7} \times 5}{\cancel{90}^{18} \times 5} = \frac{7}{18}$$ **Final answer:** $$\boxed{\frac{7}{18}}$$