1. **State the problem:** Solve the equation $$a + \frac{2}{6} - \frac{1}{a} + 2 = \frac{1}{6}$$ for $a$. 2. **Simplify constants:** Note that $\frac{2}{6} = \frac{1}{3}$, so rewrite the equation as: $$a + \frac{1}{3} - \frac{1}{a} + 2 = \frac{1}{6}$$ 3. **Combine like terms on the left:** $\frac{1}{3} + 2 = \frac{1}{3} + \frac{6}{3} = \frac{7}{3}$, so: $$a + \frac{7}{3} - \frac{1}{a} = \frac{1}{6}$$ 4. **Isolate terms:** Move constants to the right: $$a - \frac{1}{a} = \frac{1}{6} - \frac{7}{3}$$ 5. **Find common denominator on the right:** $\frac{1}{6} - \frac{7}{3} = \frac{1}{6} - \frac{14}{6} = -\frac{13}{6}$, so: $$a - \frac{1}{a} = -\frac{13}{6}$$ 6. **Multiply both sides by $a$ (assuming $a \neq 0$) to clear denominator:** $$a^2 - 1 = -\frac{13}{6}a$$ 7. **Bring all terms to one side:** $$a^2 + \frac{13}{6}a - 1 = 0$$ 8. **Multiply entire equation by 6 to clear fractions:** $$6a^2 + 13a - 6 = 0$$ 9. **Use quadratic formula:** For $6a^2 + 13a - 6 = 0$, $$a = \frac{-13 \pm \sqrt{13^2 - 4 \times 6 \times (-6)}}{2 \times 6} = \frac{-13 \pm \sqrt{169 + 144}}{12} = \frac{-13 \pm \sqrt{313}}{12}$$ 10. **Final answer:** $$a = \frac{-13 + \sqrt{313}}{12} \quad \text{or} \quad a = \frac{-13 - \sqrt{313}}{12}$$