Let $$f(x)=\sum_\limits{k=1}^{10} k x^{k}, x \in \mathbb{R}$$. If $$2 f(2)+f^{\prime}(2)=119(2)^{\mathrm{n}}+1$$ then $$\mathrm{n}$$ is equal to ___________

For $$x \in(-1,1]$$, the number of solutions of the equation $$\sin ^{-1} x=2 \tan ^{-1} x$$ is equal to __________.

If $$y=y(x)$$ is the solution of the differential equation

$$\frac{d y}{d x}+\frac{4 x}{\left(x^{2}-1\right)} y=\frac{x+2}{\left(x^{2}-1\right)^{\frac{5}{2}}}, x > 1$$ such that

$$y(2)=\frac{2}{9} \log _{e}(2+\sqrt{3}) \text { and } y(\sqrt{2})=\alpha \log _{e}(\sqrt{\alpha}+\beta)+\beta-\sqrt{\gamma}, \alpha, \beta, \gamma \in \mathbb{N} \text {, then } \alpha \beta \gamma \text { is equal to }$$ :

Let $$[\alpha]$$ denote the greatest integer $$\leq \alpha$$. Then $$[\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]+\ldots+[\sqrt{120}]$$ is equal to __________