1. Stating the problem: Evaluate the expression $$560.4 + \left\{ \left[ (3 \times 4 + 2) \times 2 - 2 \times 6 + (3 \times 6 + 1) \right] : 5 + 7 \right\} : 14 + 4$$ 2. Calculate inside the innermost brackets: $$3 \times 4 + 2 = 12 + 2 = 14$$ 3. Multiply by 2: $$14 \times 2 = 28$$ 4. Calculate the next term: $$2 \times 6 = 12$$ 5. Calculate the other bracket: $$3 \times 6 + 1 = 18 + 1 = 19$$ 6. Combine inside the square brackets: $$28 - 12 + 19 = 16 + 19 = 35$$ 7. Divide by 5: $$\frac{35}{5} = 7$$ 8. Add 7: $$7 + 7 = 14$$ 9. Divide by 14: $$\frac{14}{14} = 1$$ 10. Add 4: $$1 + 4 = 5$$ 11. Finally add 560.4: $$560.4 + 5 = 565.4$$ Final answer: $$\boxed{565.4}$$