
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
For a positive integer \( n \), let \( \theta(n) \) denote the number of integers \( 0 \leq x < 2010 \) such that \( x^2 - n \) is divisible by 2010. Determine the remainder when \( \sum_{n=0}^{2009} n \cdot \theta(n) \) is divided by 2010.

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