\[ \displaystyle \Omega_{n}(x) =\frac{\displaystyle \sum_{i=0}^n \cos \bigg[(2i+1)x \bigg]}{\displaystyle \sum_{i=0}^n \sin \bigg [ ({2i+1})x \bigg]} \]

Consider the function above, if \( \displaystyle \Omega_{200}\left(\frac{\pi}{8}\right)\) can be expressed in the form \(\sqrt{a+b\sqrt{c}}\), where \(a,b,c\) are positive integers, with \(b\) square-free.

What is the value of \( a+b+c\)?

This is my 200-follower problem. It has been a very, very interesting journey so far. Thank you Brilliant!

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