Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined as $f(x)=a\sin\left(\frac{\pi[x]}{2}\right)+[2x]$, $a\in\mathbb{R}$, where $[t]$ is the greatest integer less than or equal to $t$. If $\lim_{x\to0}f(x)$ exists, then the value of $\int_0^2 f(x)dx$ is equal to