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 $2010$.

Let's think step by step and output the final answer within \boxed{}.
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