
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 \( \mathbb{R} \) be the set of real numbers. We denote by \( \mathcal{F} \) the set of all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that
$$
f(x+f(y))=f(x)+f(y)
$$
for every \( x, y \in \mathbb{R} \). Find all rational numbers \( q \) such that for every function \( f \in \mathcal{F} \), there exists some \( z \in \mathbb{R} \) satisfying \( f(z)=q z \).

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