
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
Find all pairs of nonzero integer values \( m \leq n \) that satisfy the inequality \( m+n \neq 0 \) and the identity

\[ f_{m}(x, y) f_{n}(x, y) = f_{m+n}(x, y), \quad x, y \in \mathbf{R}, \quad x y(x+y) \neq 0 \]

where

\[ f_{k}(x, y) = \frac{x^{k} + y^{k} + (-1)^{k} (x+y)^{k}}{k} \]

Hint: The pairs \( m=2, n=3 \) and \( m=2, n=5 \) satisfy the required conditions.

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