Let's define a series \(S\), with \(0^{th}\) element \({ a }_{ 0 }\) as:

\[S({ a }_{ 0 })={ \quad \{ a }_{ 1 },{ a }_{ 2 },{ a }_{ 3 },{ a }_{ 4 },...\} \]

Where \({ a }_{ n }=\quad \varphi { a }_{ n-1 }+\frac { 1 }{ \varphi } \quad or\quad { a }_{ n }=\frac { { a }_{ n-1 } }{ \varphi } -\frac { 1 }{ { \varphi }^{ 2 } } \) with equal probability.

If \({ a }_{ 0 }=31\), the expected value of \({a}_{51}\) can be expressed as:

\[\frac { { a }^{ b }\sqrt { c } }{ { d }^{ e } } -f\]

Find: \(a+b+c+d+e+f\)

**Details and Assumptions:**

- \(\varphi=\frac { \sqrt { 5 } +1 }{ 2 } \)
- \(a,b,c,d,e,f\) are integers. \(a\) and \(d\) are co-prime. \(c\) is a square-free number.

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