
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
a) Find all distinct positive integers \( x \) and \( y \) that satisfy the equation

\[ x^{y} = y^{x} \]

b) Find all distinct positive rational numbers \( x \) and \( y \) that approximately satisfy the equation

\[ x^{y} \approx y^{x} \]

(specify the formula that gives all solutions).

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