
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, while the other die is biased such that it comes up with six with probability $\frac{1}{2}$ and each of the other five sides has probability $\frac{1}{10}$. Charles randomly selects one of the two dice and rolls it five times. Given that the first four rolls include at least two sixes, the probability that the fifth roll is also 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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