If you think something out(=to a conclusion or result), you consider it carefully before making a decision.
- I need some time to think things out.
If some people say they can't argue something out(=to a conclusion or result), it means they cannnot reach any conclusion to it.
The "out" functions as an adverb that means "to a conclusion or result", so "I'd like to think it out" sounds awkward.
If you think something through, you consider it thoroughly(=in every part or way), together with all possible effects or consequences.
- I didn't think through the consequences of refusing to take action.
Don't be so obsessed with the "through", it's as simple as this.
- I spent my summer vacation touring through(= various places in) France.
- I'd like you to think through(=in every part or way) the consequences of refusing to take action.