
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
For any integer \( r \geq 1 \), determine the smallest integer \( h(r) \geq 1 \) such that for any partition of the set \(\{1,2, \ldots, h(r)\}\) into \( r \) classes, there are integers \( a \geq 0, 1 \leq x \leq y \), such that \( a+x, a+y, a+x+y \) belong to the same class.

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