
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
The function $f(x)$ takes positive real numbers to real numbers, such that
\[ k(xf(y) - yf(x)) = f \left( \frac{x}{y} \right) \]
for all positive real numbers $x$ and $y$, where $k$ is a non-zero real constant. Find all possible values of $f(100)$.

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