
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
This year, some contestants at the Memorial Contest ABC are friends with each other (friendship is always mutual). For each contestant  $X$ , let  $t(X)$  be the total score that this contestant achieved in previous years before this contest. It is known that the following statements are true: $1)$  For any two friends  $X'$  and  $X''$ , we have  $t(X') \neq t(X''),$  $2)$  For every contestant  $X$ , the set  $\{ t(Y) : Y \text{ is a friend of } X \}$  consists of consecutive integers.
The organizers want to distribute the contestants into contest halls in such a way that no two friends are in the same hall. What is the minimal number of halls they need?

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