
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 \(\mathbb{Q}\) be the set of rational numbers. Given a rational number \(a \neq 0\), find, with proof, all functions \(f: \mathbb{Q} \rightarrow \mathbb{Q}\) satisfying the equation
$$
f(f(x)+ay)=af(y)+x
$$
for all \(x, y \in \mathbb{Q}\).

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