How to simplify the following expression through Boolean algebra

Disclaimer: This was a homework problem from the first assignment of the semester - the assignment has long since been graded. For the life of me I can't crack this one - I don't understand what I'm not seeing here.

I know from applying a Karnaugh Map to the following Boolean expression that:

$\bar{A} B \bar{C} + \bar{A} \bar{B} C + A \bar{B} \bar{C} + \bar{A} \bar{B} \bar{C} = \bar{A} \bar{C} + \bar{A} \bar{B} + \bar{B} \bar{C}$

Additionally, Wolfram Alpha computes the same simplified solution.

The most simplified version of the equation that I can get is:

$\bar{A} B \bar{C} + \bar{A} \bar{B} C + A \bar{B} \bar{C} + \bar{A} \bar{B} \bar{C}$

$\bar{A} B \bar{C} + \bar{A} \bar{B} C + \bar{B} \bar{C} (\bar{A} + A)$

$\bar{A}(B \bar{C} + \bar{B} C) + \bar{B} \bar{C} (1)$

$\bar{A}(B \oplus C) + \bar{B} \bar{C}$

$\bar{A}B \oplus \bar{A}C + \bar{B} \bar{C}$

My guess is that I'm taking a wrong turn somewhere. Any help or suggestions will be greatly appreciated.

$\bar{A} B \bar{C} + \bar{A} \bar{B} C + A \bar{B} \bar{C} + \bar{A} \bar{B} \bar{C}$
$\bar{A} B \bar{C} + \bar{A} \bar{B} \bar{C}+\bar{A} \bar{B} C + \bar{A} \bar{B} \bar{C}+ A \bar{B} \bar{C} + \bar{A} \bar{B} \bar{C}$
$\bar{A} \bar{C} (B + \bar{B}) +\bar{A} \bar{B} ( C + \bar{C}) + \bar{B} \bar{C} ( A+ \bar{A})$
$\bar{A} \bar{C} +\bar{A} \bar{B} + \bar{B} \bar{C}$.