What to do with a random chicken that randomly showed up at my house?

A random hen showed up at my house about three weeks ago and I'm unsure what to do with her. None of my neighbors have chickens so she's not theirs and I haven't seen anyone else around with them either. My yard is 1 acre and surrounded by woods and bamboo. She stays along the perimeter of the woods and will come within ten feet of the house front door where I have bird feeders to eat the seed on the ground.

I bought some chicken food and have been leaving her piles out under the bird feeder and along where I see her most often next to the woods. She's skittish and will run away cackling if we startle her but won't run completely away, just far enough that no one could get her and then stare at you. She doesn't trust anyone enough to come near if you call her and shake a cup of food but she will stay just out of reach and come eat the food when I go back inside.

Should I try to catch her, build a coop and see if she sleeps in it? I'm worried about her sleeping in the bamboo and want to do something for her.