
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 an upcoming international mathematics contest, the participating countries were asked to choose from nine combinatorics problems. Given how hard it usually is to agree, nobody was surprised that the following happened:**i)** Every country voted for exactly three problems.**ii)** Any two countries voted for different sets of problems.**iii)** Given any three countries, there was a problem none of them voted for.
Find the maximal possible number of participating countries.

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