Counting Values of a Sum of Floor Functions

For a real number $t$, $\lfloor t \rfloor$ is the greatest integer that is less than or equal to $t$. Find the number of possible values the expression $$\left\lfloor \frac{x}{3} \right\rfloor + \left\lfloor \frac{x}{5} \right\rfloor + \left\lfloor \frac{x}{7} \right\rfloor + \left\lfloor \frac{x}{8} \right\rfloor$$ can take over all real numbers $x \in [0,840]$.