Engine Starting Problems w/87 5 speed V6 2.8 L

I have a 1987 Chev Camaro V6 5 speed manual. When I start my engine after letting it sit for a day or so, it starts on the first try like normal. However, when I start it for the 2nd and so on times, it does
not want to start, and it takes like 3 tries before it decides to kick on. After a lot of ignitions in one day, it stops turning altogether. I know it is not the battery, and my alternator is good because once I push start it, it goes on until the gas runs out. What are the possible parts which are causing this problem?
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