
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
Let  $n \geq 2$  be a positive integer. Determine the number of  $n$ -tuples  $(x_1, x_2, \ldots, x_n)$  such that  $x_k \in \{0, 1, 2\}$  for  $1 \leq k \leq n$  and  $\sum_{k = 1}^n x_k - \prod_{k = 1}^n x_k$  is divisible by  $3$ .

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