
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 \minus{} n$  is divisible by  $ 2010$ . Determine the remainder when  $ \displaystyle \sum_{n \equal{} 0}^{2009} n \cdot \theta(n)$  is divided by  $ 2010$ .

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