In a laboratory, researchers use two different chemical catalysts, $C_1$ and $C_2$, to trigger a reaction. Catalyst $C_1$ is used $65\%$ of the time and has a $12\%$ failure rate, while $C_2$ is used $35\%$ of the time and has an $8\%$ failure rate. If a reaction succeeds (meaning it did not fail), what is the probability that catalyst $C_2$ was used?