They are not mutually exclusive. The united states is a democracy.
"Republic" is just a term to refer to a nation that is ruled without a monarch: https://www.merriam-webster.com/dictionary/republic
Therefore, the two are not mutually exclusive.
If you actually meant a direct democracy, then you should have said it. Don't expect people to be able to read your mind and then get mad and snarky when they don't. We can only interpret what we read, so being specific that you mean a direct democracy when you say the word "democracy" in a political thread is really needed.
We have given you all the actual definitions on these terms and they show clearly that you're wrong. Just admit it and stop arguing this pointless debate. It's not that hard to do.