
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 one six-sided fair die, one eight-sided biased die, and one ten-sided biased die. The eight-sided biased die comes up eight with probability $\frac{3}{4}$ and each of the other seven sides with a probability of $\frac{1}{28}$. The ten-sided biased die comes up ten with probability $\frac{4}{5}$ and each of the other nine sides with a probability of $\frac{1}{45}$. Charles randomly chooses one of the three dice and rolls it four times. Given that the first three rolls are all showing the maximum face value of the respective dice (i.e., six, eight, or ten), the probability that the fourth roll will also show the maximum face value 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}
user