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#Yes, you can force them to tell you or relay things to you

You can use your "total and precise control" to simply force them to relay to you the information you need.

Until the end of your next turn, the creature takes only the actions you choose, and doesn't do anything that you don't allow it to do.

If there are concerns about speaking not technically being an action you can tell them for example, to write down the answer to your question on a piece of paper which would be Use an Object likely.

Combined with a simple command of "tell the truth" you could get any information you needed out of them.

###You don't get access to their thoughts or memories

The only sentence that could possibly be construed as giving that would be:

You can use your action to take total and precise control of the target.

However, the spell explains in the next line what this sentence means by "total and precise control":

Until the end of your next turn, the creature takes only the actions you choose, and doesn't do anything that you don't allow it to do.

Nowhere does it say anything about granting you insight into their thoughts or memories so it does not allow you to do so. Spells only do what they say they do and the only thing this spell does is give you puppet like control of their actions.