1. **State the problem:** Simplify the expression $$2 \left(\frac{1}{2}\right) \div \left(\frac{5}{6} - \frac{1}{3}\right) + 1$$. 2. **Rewrite the expression:** $$2 \times \frac{1}{2} \div \left(\frac{5}{6} - \frac{1}{3}\right) + 1$$. 3. **Simplify multiplication:** $$2 \times \frac{1}{2} = \frac{2}{1} \times \frac{1}{2} = \frac{2 \times 1}{1 \times 2} = \frac{2}{2} = 1$$. 4. **Simplify the subtraction inside the parentheses:** Find a common denominator for $$\frac{5}{6}$$ and $$\frac{1}{3}$$, which is 6. $$\frac{1}{3} = \frac{2}{6}$$. So, $$\frac{5}{6} - \frac{2}{6} = \frac{5 - 2}{6} = \frac{3}{6}$$. 5. **Simplify the fraction:** $$\frac{3}{6} = \frac{\cancel{3}}{\cancel{6}} = \frac{1}{2}$$. 6. **Rewrite the expression now:** $$1 \div \frac{1}{2} + 1$$. 7. **Division by a fraction is multiplication by its reciprocal:** $$1 \div \frac{1}{2} = 1 \times \frac{2}{1} = 2$$. 8. **Add 1:** $$2 + 1 = 3$$. **Final answer:** $$3$$.