
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
Charles has two six-sided dice. One of the dice is fair, and the other die is biased so that it comes up six with probability $\frac{1}{2}$ and five with probability $\frac{1}{6}$, while each of the other four sides has a probability of $\frac{1}{24}$. Charles chooses one of the dice randomly and rolls it three times. Given that the first two rolls show either sixes or fives, the probability that the third roll will also show a six or five is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

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