
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 every integer  $r > 1$  find the smallest integer  $h(r) > 1$  having the following property: For any partition of the set  $\{1, 2, . . ., h(r)\}$  into  $r$  classes, there exist integers  $a \geq 0, 1 \leq x \leq y$  such that the numbers  $a + x, a + y, a + x + y$  are contained in the same class of the partition.

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