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