I want to prove that for every short exact sequence $$ O \to A \xrightarrow{f} B \xrightarrow{g} C \to O $$ of $R$-module homomorphisms, if the induced sequence $$ O \to M\otimes_R A \xrightarrow{\operatorname{id}_M\otimes f} M\otimes_R B \xrightarrow{\operatorname{id}_M\otimes g} M\otimes_R C \to O $$ is exact, then $M$ is flat over $R$.
I know that $M$ is flat if given a short exact sequence, the induced sequence is exact. However, the question I am asking here goes in the reverse direction, and I don't know how to go about it. I will appreciate your help.
Thanks.