Boolean expression simplification

I know this stuff should be way easy once I figure it out, but right now I am literally pulling out my hair trying to do so.(Headbang) The original expression is:

A`D` + A`BC + A` + AB` + A`BC + ABC`

I see that A and A` are common factors, so I pull them out:

A`(D` + BC + 1 + BC) + A(B` + BC`)

The fact that there is a "1" in the first parentheses makes everything in the parentheses 1, right?

A`(1) + A(B` + BC`)

X and 1 = X

A` + A(B` + BC`)

The only thing I can think to do is distribute the (B` + BC`) but I don't know if that is the right thing to do...

A` + A(B+B`)(B`+C`)

Now, (B+B`) is equal to 1. A and 1 is equal A.

A` + A(B`+C`)

Now I am stuck. I have no way to check my answer, but Wolfram says the simplified answer is:

A`+B`+C`

Can you help me finish/correct the above work?