Sum of all real values of $x$ satisfying the equation ${\log }_p\left({\log }_4\left|x\right|-2+p^x\right)=x, p>2$ is

Let $\left[a\right]$ denote the greatest integer less than or equal to '$a$'. The set of all values of  '$x$' satisfying the inequality,

$\left|2x^2-4x-7\right|<\left[1+\frac{1}{2}{\left(\frac{\cos \theta }{\cos \frac{\theta }{2}-\sin \frac{\theta }{2}}\right)}^2\right], -\frac{\pi }{2}\le \theta <\frac{\pi }{2}$ is

$S_1:\left[x+\frac{1}{2}\right]=\left[x\right]$ if $\left\{x\right\}<\frac{1}{2}$

$S_2:\left[nx\right]=n\left[x\right]$ if $\left\{x\right\}<\frac{1}{n}$

where, $\left[·\right]$ and $\left\{·\right\}$  stands for greatest integer and fraction part functions respectively and $n$ is a natural number

If for all positive integers $n,$ we are given that $a_n=n^2+20, n\in N$ and $d_n$ is the highest common factor ($\mathrm{H}.\mathrm{C}.\mathrm{F}.$) of $a_{n+1}$ and $a_n,$ then the value of $\left\{\frac{2n+1}{d_n}\right\}$ is (where $\{.\}$ denotes fractional part function)