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