
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
Given the function
$$
f_{k}(x, y)=\frac{x^{k}+y^{k}+(-1)^{k}(x+y)^{k}}{k}, \quad k \in \mathbb{Z}, \; k \neq 0,
$$

find all non-zero integer pairs \((m, n)\) such that \(m \leqslant n, m+n \neq 0\), and
$$
f_{m}(x, y) f_{n}(x, y) \equiv f_{m+n}(x, y),
$$

where \( x, y \in \mathbb{R}, \; x y(x+y) \neq 0 \).

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

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