
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
Electoral ballots. A certain physical society needed to hold elections for three leadership positions. For each of the three positions, there were 3, 4, and 5 candidates respectively. To ensure that the number under which each candidate was listed on the ballot did not influence the election results, it was decided to use a rule that would require each candidate to appear under each number an equal number of times on the ballot for their respective position. What is the minimum number of different ballots required to meet this rule?

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