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Had i caught one, I would not approach the person or try to restrain the person without a partner.

I would order him to knee down on the ground, feet crossed and hands behind the head. I would tell him any sudden movement would be treated as hostile attack to me.

If he runs away, that's OK and I would be a good witness.

In your scenario, if they enter the yard, they get to be told to leave (trespassing) without showing the firearm; if they break into the house (home invasion), they get shot first.