$x ≡ a \pmod{m}, x ≡ b\pmod{n}, x ≡ c\pmod{r}$

Question

What could be an example of three positive integers $m, n$, and $r$, and three integers $a, b$, and $c$ such that the $\mathrm{gcd}$ of $m, n$, and $r$ is $1$, but there is no simultaneous solution to

$x ≡ a \pmod{m}$

$x ≡ b\pmod{n}$

$x ≡ c\pmod{r}$