
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
Suppose the function $g(x)$ maps positive real numbers to real numbers, satisfying the relation
\[ xg(y) - yg(x) = g\left( \frac{x}{y} \right) + x - y \]
for all positive real numbers $x$ and $y.$ Determine all possible values of $g(100).$

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