
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
There are three bags. One bag contains three green candies and one red candy. One bag contains two green candies and two red candies. One bag contains one green candy and three red candies. A child randomly selects one of the bags, randomly chooses a first candy from that bag, and eats the candy. If the first candy had been green, the child randomly chooses one of the other two bags and randomly selects a second candy from that bag. If the first candy had been red, the child randomly selects a second candy from the same bag as the first candy. If the probability that the second candy is green is given by the fraction  $m/n$  in lowest terms, find  $m + n$ .

Present the answer in LaTex format: \boxed{Your answer}
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