Dual US-Taiwanese citizenship

I was born in the States to Taiwanese parents. I had a Taiwanese passport as a baby but it expired long ago. I have acquired a new Taiwanese passport but it doesn’t qualify me for citizenship. From what I understand, I can stay for three months at a time, with a one-time extension. I assume this means that every three months I have to make a visa run until I have lived here for one year, and then I can apply for citizenship. If this is wrong, or if anyone is in the same situation, I’d love to know about it. Thanks!

My case differs from you in many ways.
I have two passports, one USA and one ROC.
I am in the process of applying for the ID that everybody carries.
On my ROC passport, there is an ID number.
I think this is the number you need to apply for some sort of legal residence here.
Do you have such ID number?