
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
Find the number of ordered 2012-tuples of integers \(\left(x_{1}, x_{2}, \ldots, x_{2012}\right)\), with each integer between 0 and 2011 inclusive, such that the sum \(x_{1}+2 x_{2}+3 x_{3}+\cdots+2012 x_{2012}\) is divisible by 2012.

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