1. Stating the problem: Calculate the value of the expression $$\frac{\left(\frac{3}{7} + \frac{1}{2}\right)^2 \cdot \left(2 - \frac{1}{3}\right)^2 - 1}{\left(\frac{1}{6} - 1\right)^2}$$ 2. Calculate each part step-by-step. 3. Sum inside the first parentheses: $$\frac{3}{7} + \frac{1}{2} = \frac{3 \cdot 2}{7 \cdot 2} + \frac{1 \cdot 7}{2 \cdot 7} = \frac{6}{14} + \frac{7}{14} = \frac{13}{14}$$ 4. Square the sum: $$\left(\frac{13}{14}\right)^2 = \frac{169}{196}$$ 5. Calculate inside the second parentheses: $$2 - \frac{1}{3} = \frac{6}{3} - \frac{1}{3} = \frac{5}{3}$$ 6. Square the difference: $$\left(\frac{5}{3}\right)^2 = \frac{25}{9}$$ 7. Multiply the squared terms: $$\frac{169}{\cancel{196}} \cdot \frac{25}{\cancel{9}} = \frac{169 \cdot 25}{196 \cdot 9} = \frac{4225}{1764}$$ 8. Subtract 1: $$\frac{4225}{1764} - 1 = \frac{4225}{1764} - \frac{1764}{1764} = \frac{4225 - 1764}{1764} = \frac{2461}{1764}$$ 9. Calculate denominator: $$\frac{1}{6} - 1 = \frac{1}{6} - \frac{6}{6} = -\frac{5}{6}$$ 10. Square denominator: $$\left(-\frac{5}{6}\right)^2 = \frac{25}{36}$$ 11. Divide numerator by denominator: $$\frac{\frac{2461}{1764}}{\frac{25}{36}} = \frac{2461}{1764} \cdot \frac{36}{25} = \frac{2461 \cdot 36}{1764 \cdot 25}$$ 12. Simplify fraction by canceling common factors: $$\frac{2461 \cdot \cancel{36}}{\cancel{1764} \cdot 25} = \frac{2461 \cdot 36}{1764 \cdot 25}$$ Note: 1764 = 36 \times 49, so cancel 36: $$= \frac{2461}{49 \cdot 25} = \frac{2461}{1225}$$ 13. Final answer: $$\boxed{\frac{2461}{1225}}$$ This is the simplified exact value of the expression.