
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
Evan is studying the occurrence of three genetic traits, denoted by X, Y, and Z, within a population of women. The probability that a randomly selected woman solely has trait X (and none of the others) is 0.12. For each of the other individual traits Y and Z, the probability is 0.11. The probability that a randomly selected woman has exactly two out of these three traits, but not the third one, is consistently 0.15 for each pair. Furthermore, the probability that a woman has all three traits Y, Z, and X, given that she has X and Y, is $\frac{1}{5}$. The probability that a woman does not have any of the three traits given that she lacks trait X is still expressed as $\frac{p}{q}$ where $p$ and $q$ are relatively prime integers. Find $p+q$.

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