
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
Given the equation \( y^x = x^y \), answer the following questions:

a. What is the set of positive numbers \( x \) for which the only positive solution \( y \) of this equation is the trivial solution \( y = x \)?

b. For those values of \( x \) for which non-trivial solutions \( y \) exist for this equation, how many such non-trivial solutions are there for each \( x \)?

c. If we randomly select a value \( x \) from the interval \( (0, e) \), what is the probability that the corresponding non-trivial solution \( y \) will lie in the same interval?

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