I once heard someone state that Imam Ali once said (I am heavily paraphrasing), that if one is in dispute over a matter, to pick one side, and to stick with that one side.

Could you clarify the exact wording and source of this saying, and which context it was used and for what purpose?

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I am not aware of such a hadith. In general, the Imams have prohibited us from being excessive in disputes and arguing too much. We should state the truth and not get stuck in arguing.