1 over $\pi$ and $\phi$

$\lim_{m\to \infty^+}\lim_{n\to\infty^+}\prod_{k=1}^{m}\operatorname{\cos }\left(2\pi \sqrt[2^k]{ n^{2^k}+ \frac{n^{2^k-1}}{120}+ 2^k}\right)$ For $n$ being positive integer and the value of the above expression can be expressed $\frac{a}{\phi\pi}\left(\frac{1}{{\sqrt b}}+\sqrt {\frac{c}{2}} -\frac{\phi}{\sqrt{2}}\left(\sqrt 3-1\right)\sqrt{1+\phi^2}\right)$ where $a$ , $b$ and $c$ are positive integers with $b$ and $c$ being primes. Find the value $a+b+c$ .

Notation: $\phi=2^{-1}\left(1+\sqrt 5\right)$ denotes Golden ratio

Inspired by (1) and (2) .