
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
Let the function \( g(x) \) take positive real numbers to real numbers such that
\[ xg(y) - yg(x) = g \left( \frac{x}{y} \right) + x - y \]
for all positive real numbers \( x \) and \( y \). Find all possible values of \( g(50) \).

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