Define

\[f(x) = \dfrac{1}{2} \tan\left(\dfrac{x}{2}\right) + \dfrac{1}{4} \tan\left(\dfrac{x}{4}\right) + \dfrac{1}{8} \tan\left(\frac{x}{8}\right) +\cdots.\]

Then \(f\left(\dfrac{3\pi}{8}\right) \) can be written as \[ \dfrac{a}{b\pi^{c}} + d - \sqrt{e}\]

If \(\dfrac{a}{b}\) is in simplest form i.e. \(\gcd(a,b)=1\), and \(a,b,c,d,e\in\mathbb N\), find the value of \(a + b + c + d + e\).

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