
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
Consider the set  $A = \{1, 2, 3, ..., 2008\}$ . We say that a set is of *type*  $r, r \in \{0, 1, 2\}$ , if that set is a nonempty subset of  $A$  and the sum of its elements gives the remainder  $r$  when divided by  $3$ . Denote by  $X_r, r \in \{0, 1, 2\}$  the class of sets of type  $r$ . Determine which of the classes  $X_r, r \in \{0, 1, 2\}$ ,  is the largest.

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