1. **State the problem:** Simplify the expression $$\frac{1}{2} + \frac{1}{2 + \frac{12}{7}}$$. 2. **Rewrite the denominator in the second term:** The denominator is $$2 + \frac{12}{7}$$. 3. **Find a common denominator for the sum in the denominator:** $$2 = \frac{14}{7}$$, so $$2 + \frac{12}{7} = \frac{14}{7} + \frac{12}{7} = \frac{26}{7}$$. 4. **Rewrite the expression:** $$\frac{1}{2} + \frac{1}{\frac{26}{7}}$$. 5. **Simplify the second term by dividing by a fraction:** $$\frac{1}{\frac{26}{7}} = \frac{7}{26}$$. 6. **Now the expression is:** $$\frac{1}{2} + \frac{7}{26}$$. 7. **Find a common denominator to add the fractions:** The least common denominator of 2 and 26 is 26. 8. **Rewrite each fraction with denominator 26:** $$\frac{1}{2} = \frac{13}{26}$$. 9. **Add the fractions:** $$\frac{13}{26} + \frac{7}{26} = \frac{20}{26}$$. 10. **Simplify the fraction by dividing numerator and denominator by their greatest common divisor 2:** $$\frac{20}{26} = \frac{\cancel{20}^{10}}{\cancel{26}^{13}}$$. **Final answer:** $$\boxed{\frac{10}{13}}$$