
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
The set of points with integer coordinates is referred to an orthonormal coordinate system \((0, \overrightarrow{1}, \vec{\jmath})\). Let \(\mathcal{E}\) be the set of points with integer coordinates. Find all functions \(f: \mathcal{E} \rightarrow \mathbb{R}\) such that \(f(O) = 0\) and for every rectangle \(OABC\) with \(A, B, C \in \mathcal{E}\), the equation \(f(B) = f(A) + f(C)\) holds.

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