
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 three six-sided dice. One die is fair, the second die is biased so that it comes up six with probability $\frac{2}{3}$ and each of the other five sides has probability $\frac{1}{15}$, and the third die is biased so that it comes up six with probability $\frac{1}{3}$ and each of the other five sides has probability $\frac{2}{15}$. Charles chooses one of the dice at random and rolls it four times. Given that the first three rolls are all sixes, calculate the probability that the fourth roll will also be a six as $\frac{p}{q}$, 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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