
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  $ M \ge 3$  be an integer and let  $ S \equal{} \{3,4,5,\ldots,m\}$ . Find the smallest value of  $ m$  such that for every partition of  $ S$  into two subsets, at least one of the subsets contains integers  $ a$ ,  $ b$ , and  $ c$  (not necessarily distinct) such that  $ ab \equal{} c$ .**Note**: a partition of  $ S$  is a pair of sets  $ A$ ,  $ B$  such that  $ A \cap B \equal{} \emptyset$ ,  $ A \cup B \equal{} S$ .

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