
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
Determine the smallest integer $k \geq 2$ that satisfies the following property. For any partition of the set $\{2,3, \ldots, k\}$ into two parts, at least one of the parts contains three numbers $a, b, c$, not necessarily distinct, such that $a b = c$.

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