Solving $xy+z=a$, $x+y=b$, $yz=c$. Is there a trick to avoid a cubic?

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How may one solve this system?

$$\begin{align} xy+z &=a \\ x+y &=b \\ yz &=c \end{align}$$

It keeps leading to a cubic in one of the variables when you apply direct substitution, which is tedious to solve. Is there a way to circumvent this? Some trick?

Many thanks.