For positive integers $n$, let the numbers $c(n)$ be determined by the rules:  
- $c(1) = 1$  
- $c(2n) = c(n)$  
- $c(2n+1) = (-1)^n c(n)$  

Find the value of \( \sum_{n=1}^{2013} c(n) c(n+2) \).

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