Given \[f(x)=\frac{\sqrt{3x^2+1}}{x}.\] We define a positive algebraic function \(g\) such that the composition \(fog\) satisfies the equation \[\prod_{r=1}^n (fog(r))^2=^{n+3}P_n\] for every positive integer \(n\). Find \(4g(4)\).

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