Drawing Marbles #2

The content of 3 bags w.r.t green and red marbles is as given in the table shown.

$\begin{array}{l|c|r}Bag&G&R\\ \hline A&3&1\\ \hline B&2&2\\ \hline C&1&3\end{array}$

A child randomly selects one of the bags, draws a marble from it and retains it. If the first marble is green,the child draws a second marble randomly from one of the two remaining bags. If the first marble drawn is red the child draws one more marble from the same bag. The probability that the second drawn marble is green is expressed as $\frac{m}{n}$ (where m and n are coprime). Find the value of $m+n$