Consider the function \( f:\mathbb{R}\to\mathbb{R} \) defined by \( f(x)=\dfrac{2x}{\sqrt{1+9x^{2}}} \). If the composition of \( (\underbrace{f \circ f \circ f \circ \dots \circ f}_{10\text{ times}})(x) = \frac{2^{10}x}{\sqrt{1 + 9\alpha x^2}} \), then the value of \( \sqrt{3\alpha+1} \) is equal to __________.
