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Recently, I texted her to ask her to go out, but she explained to me that she is in deep trouble in her life right now, that she would get back to me as soon as she were capable.
Secretive and the "Get back as soon as capable": BIG RED FLAG. Unspecified and vague time period.
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So I told her to take her time, that it was fine (that was a week ago) and have not texted her ever since.
OP told her to take her time and that it was fine. This statement is likely to be used against him when he starts the discussion about ending the relationship. If the girl doesn't have any options, she will try to keep him even if she is not that interested.
So that is one week without any contact? Then whenever she remembers to re-initiate does that give her the right to continue again from the exact same point where you left, just as simply like pressing the pause button on a media player? It does not work this way. If she was interested she would have sent any message to you within this ONE WHOLE WEEK in this supposedly relationship.
My feeling is that even if you talk to her about how you feel and even if she re-assures you with words that she will put an effort down the line, at the end of the day she is not that interested anymore, she is not willing to do as she promises and you are probably wasting your time. Just my 2 cents.