
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
Charlotte has three six-sided dice. One die is fair, one is biased so that it rolls a six with a probability of $\frac{1}{2}$, and the other five sides with equal probabilities, and the third die is biased so that it rolls a one with a probability of $\frac{1}{2}$, other sides with equal probabilities. Charlotte randomly picks one of the dice and rolls it four times. Given that the first three rolls result in sixes, the probability that the fourth roll will also be a six 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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