Let \( f:(1,\infty)\to \mathbb{R} \) be a function defined as \( f(x)=\dfrac{x-1}{x+1} \). Let \( f^{\,i+1}(x)=f\!\left(f^{\,i}(x)\right),\ i=1,2,\ldots,25 \), where \( f^{1}(x)=f(x) \). If \( g(x)+f^{26}(x)=0,\ x\in(1,\infty) \), then the area of the region bounded by the curves \( y=g(x) \), \( 2y=2x-3 \), \( y=0 \) and \( x=4 \) is:
