Given that [tex]$x^{\frac{1}{x}} = 7$, x \in \mathbb{R}^+[/tex], find the value of

[tex]\[ P = \frac{8\left(7^x\right) + \left(23^x \cdot x\right)^{\frac{1}{x}} + x^2}{322 + 2 x^2 + 16\left(7^x\right)}. \][/tex]