
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 \(\mathcal{S}\) be the set of all points in the plane. Find all functions \(f: \mathcal{S} \rightarrow \mathbb{R}\) such that for all nondegenerate triangles \(ABC\) with orthocenter \(H\), if \(f(A) \leq f(B) \leq f(C)\), then \( f(A) + f(C) = f(B) + f(H). \)

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