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If you think in terms of probability, there is no conflict. And that is also why "never do this if you want to get that" is often a necessary, but tautologically inaccurate, simplification.

The best thing is to do both. Work at a top company, and have a side project that you can discuss with hiring managers.

My experience is identical to yours. Both my first programming gig as well as subsequent advancements has been a result of my side projects.

First job or subsequent jobs?

Both are correct