\[ \mathscr{E} = \displaystyle \sum_{n=0}^{\infty} \dfrac{\sin \left( \dfrac{\pi n}{2} + \dfrac{\pi}{4}\right) }{2^n} \]

If \( \mathscr{E} \) can be represented in the form \( \dfrac{a \sqrt{b}}{c} \), with \( a, c \) coprime positive integers and \( b\) square free, find \( a + b + c \).

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