Brainstorming cosine series

Geometry Level 4

\[\large \sum_{k=1}^{1007} \left(\cos\left(\dfrac{\pi{k}}{1007}\right)\right)^{2014}\]

If the value of the sum above is of the form \(\dfrac{a+b\dbinom{a}{b}}{2^a}\), where \(a=2b\), find \(a+b\).

\(\)
Notation: \( \dbinom MN = \dfrac {M!}{N! (M-N)!}\) denotes the binomial coefficient.

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