
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 functions \( f: \mathbb{Q} \rightarrow \mathbb{Q} \) such that
\[
\begin{aligned}
f(x) f(y) & = f(x) + f(y) - f(x y), \\
1 + f(x + y) & = f(x y) + f(x) f(y)
\end{aligned}
\]
for all rational numbers \( x, y \).

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