How to strike a friendly balance with a female coworker?

She has some kind of atitude... earlier when we met i tried to be friendly with her as in tried to smile or greet her whenever she met although she is 3 years younger than me.
But the thing is she just reciprocated she never did these things from her own which i felt bad about.
So I totally started ignoring her. I never smiled ot greeted her. Looked down/somewhere else whenever we met or passed by.
Now it has become really odd and weird ro even work together.
Both act as if there's some kind of " cold war" between us. Can't stand next ro eachother or talk even for work.
Let me mention that i am generally shy introvert person and don't really open up with girls anyway.

What's the solution?
How to strike a friendly balance with a female coworker?
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