modular arithmetic - Is it possible to solve this congruency system?

I need help to determine if this congruency system can be solved and if it can be solved how do I do it:

$$\begin{cases}x\equiv2\text{ (mod $3$)}\\ x\equiv4\text{ (mod $6$)}\\ \end{cases}$$

I do know that from the system I obtain the following:

$$\begin{align} x\equiv2\text{ (mod $3$)}\\ x\equiv4\text{ (mod $2$)}\\ x\equiv4\text{ (mod $3$)}\\ \end{align}$$

I do not know what to conclude from here. I think this system doesn't have solution, but if it is so how do I prove it.