
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 the following conditions:

- Every country voted for exactly three problems.
- Any two countries voted for different sets of problems.
- 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}
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