
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  $\{1, 2, \cdots, h(r)\}$  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.

*Proposed by Romania*

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