Til the last breath

$\large \displaystyle \sum_{n=1}^{\infty} \dfrac{n(n+2)}{1\cdot2\cdot3+2\cdot3\cdot4+\cdots +n(n+1)(n+2)}$

If the above expression is in the form $\color{#D61F06}{ \dfrac{A}{B}}$ where $\color{#D61F06}{A}$ and $\color{#D61F06}{B}$ are co-prime positive integers, then let $\color{#456461}{\phi}=\color{#D61F06}{A}\times \color{#D61F06}{B}$.

$\large \color{#69047E}{\varphi}=\sqrt{\color{#456461}{\phi}\sqrt{\color{#456461}{\phi}^2\sqrt{\color{#456461}{\phi}^3\sqrt{\color{#456461}{\phi}^4\cdots}}}}$

Find the value of $\color{#3D99F6}{\left(\dfrac{\sqrt{\color{#69047E}{\varphi}}}{5}\right)!}$.

Notation:

$!$ denotes the factorial notation. For example, $10! = 1\times2\times3\times\cdots\times10$.

Inspired by Rohit Udaiwal.

Try the Part-2 here