1. Stated problem: Calculate the value of $$7 \cdot \left(\frac{7}{2} + \frac{1}{3}\right) \cdot \frac{35}{42} + \frac{7}{42}$$. 2. First, simplify inside the parentheses: $$\frac{7}{2} + \frac{1}{3}$$. 3. Find a common denominator for $$\frac{7}{2}$$ and $$\frac{1}{3}$$, which is 6. 4. Convert fractions: $$\frac{7}{2} = \frac{21}{6}$$ and $$\frac{1}{3} = \frac{2}{6}$$. 5. Add the fractions: $$\frac{21}{6} + \frac{2}{6} = \frac{23}{6}$$. 6. Now multiply by 7: $$7 \cdot \frac{23}{6} = \frac{161}{6}$$. 7. Multiply by $$\frac{35}{42}$$: $$\frac{161}{6} \cdot \frac{35}{42} = \frac{161 \cdot 35}{6 \cdot 42}$$. 8. Simplify denominator: $$6 \cdot 42 = 252$$. 9. Calculate numerator: $$161 \cdot 35 = 5635$$. 10. So the product is $$\frac{5635}{252}$$. 11. Add $$\frac{7}{42}$$ to the result. Convert $$\frac{7}{42}$$ to denominator 252: $$\frac{7}{42} = \frac{42}{252}$$. 12. Add fractions: $$\frac{5635}{252} + \frac{42}{252} = \frac{5677}{252}$$. 13. Simplify fraction if possible. Both numerator and denominator divisible by 7: $$\frac{5677 \div 7}{252 \div 7} = \frac{811}{36}$$. 14. Final answer: $$\frac{811}{36}$$ or approximately 22.53. This completes the calculation.