Find the value of the following expression.
$$\frac{{1\frac{2}{3} \div \frac{5}{6} \times 6 + \frac{4}{5} \times \frac{1}{2} + \frac{2}{3}}}{{2 - \left[ {1\frac{1}{3} \times \left( { - \frac{3}{5}} \right) - 6\left\{ {\frac{3}{5} - \left( {3 - \frac{3}{{10}}} \right)} \right\}} \right]}}$$

Solution (By Examveda Team)

$$\eqalign{ & \frac{{1\frac{2}{3} \div \frac{5}{6} \times 6 + \frac{4}{5} \times \frac{1}{2} + \frac{2}{3}}}{{2 - \left[ {1\frac{1}{3} \times \left( { - \frac{3}{5}} \right) - 6\left\{ {\frac{3}{5} - \left( {3 - \frac{3}{{10}}} \right)} \right\}} \right]}} \cr & = \frac{{\frac{5}{3} \times \frac{6}{5} \times 6 + \frac{4}{{10}} + \frac{2}{3}}}{{2 - \left[ {\frac{4}{3} \times \frac{{ - 3}}{5} - 6\left\{ {\frac{3}{5} - \frac{{27}}{{10}}} \right\}} \right]}} \cr & = \frac{{12 + \frac{{6 + 10}}{{15}}}}{{2 - \left( { - \frac{4}{5} + \frac{{21 \times 6}}{{10}}} \right)}} \cr & = \frac{{12 + \frac{{16}}{{15}}}}{{2 - \left( { - \frac{4}{5} + \frac{{63}}{5}} \right)}} \cr & = \frac{{\frac{{196}}{{15}}}}{{2 - \frac{{59}}{5}}} \cr & = \frac{{196}}{{15}} \times \frac{{ - 5}}{{49}} \cr & = \boxed{ - \frac{4}{3}} \cr} $$