
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
A doubles tournament is arranged as follows. Each player belongs to one or two doubles. Two doubles play at most one match against each other, but if two doubles share a player, then they do not compete against each other. Given a set \(\{a_1, a_2, \ldots, a_n\}\) of distinct positive integers, what is the smallest number of players for which a tournament could be arranged such that (1) the number of matches played by each player is one of the \(a_i\), and (2) given any \(a_i\) we can find at least one player who played (exactly) that number of matches.

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