Should I reach out to an ex after a few months of no contact to wish her a happy birthday? I'm the one who called it off, but she distanced herself?

I've been doing some research on whether or not I should be reaching out to my ex for her upcoming birthday later this month. Its been 2 months since we spoke last and I've done a lot of healing since. Lots of forums say not to reach out because she's the one who didn't want this relationship in the end, but the situation is a bit complex. Yes, in the final day of seeing her I was the one who ended it verbally, but the reality was that she'd been leaving me in the dark and not communicating with me for a few weeks prior. I'd asked her what she wanted out of this relationship and if she'd be willing to try to continue working on our issues but she told me she didn't know what she wanted and that the same talks kept happening... hence why I ended it. I care about this girl very much still, which is why I think I should reach out (especially after a 5 year relationship). I just don't want to make the wrong decision having never been in this situation. AKA I ended it, but her actions led me to doing something I didn't want to. Any advice?
Should I reach out to an ex after a few months of no contact to wish her a happy birthday? I'm the one who called it off, but she distanced herself?
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