Sequence and Series: JEE Main 2024 Mathematics question

Let \alpha=\sum_\limits{r=0}^n\left(4 r^2+2 r+1\right){ }^n C_r and \beta=\left(\sum_\limits{r=0}^n \frac{{ }^n C_r}{r+1}\right)+\frac{1}{n+1}. If $140<\frac{2 \alpha}{\beta}<281$, then the value of $n$ is _________.