\[\large \displaystyle \sum_{n=1}^{\infty} \frac{\lfloor \sqrt{n} \rfloor }{\lfloor \sqrt[4]{n} \rfloor ^7 } \]

The above sum can be expressed as :

\[\displaystyle a\zeta (\alpha) + b\zeta (\beta) + \frac{c}{d} \zeta (\gamma) + e \zeta (\delta) +\frac{f}{g} \zeta (\epsilon) \]

Where \( a,b,c,d,e,f,g,\alpha,\beta,\gamma,\delta, \) and \(\epsilon \) are positive integers with \( c \) and \(d\) being coprime, \( f\) and \( g \) are also coprime.

Find : \( a+b+c+d+e+f+g+\alpha +\beta +\gamma +\delta +\epsilon \).

**Details and Assumptions**:

\(\bullet \lfloor x \rfloor \) Denotes the Floor Function.

\(\bullet \zeta ( x) \) is the Riemann Zeta Function.

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