How to prove re-hard or co-re hard by reduction?

My question asks to show that $ FINITE_{TM}$ is r.e. hard and co.re hard by reduction. My first idea is that we can show $ HALT_{TM}$ that’s reducible to it. So, my assumption is to prove $ FINITE_{TM}$ is r.e. hard, we can reduce $ HALT_{TM}$ to $ FINITE_{TM}$ . And to show $ FINITE_{TM}$ is co-r.e. hard, we can reduce $ \overline{HALT}_{TM}$ to $ FINITE_{TM}$ . Does my assumption sound accurate? Any suggestions?