
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
In the expansion of \((1+x)^n\), let \(T_r(n)\) be the number of coefficients that give a remainder of \(r\) when divided by 3, where \(r \in \{0,1,2\}\). Calculate \(T_0(2006)\), \(T_1(2006)\), and \(T_2(2006)\).

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