In \triangle A B C, the sides have integers lengths and A B=A C. Circle \omega has its center at the incenter of \triangle A B C. An excircle of \triangle A B C is a circle in the exterior of \triangle A B C that is tangent to one side of the triangle and tangent to the extensions of the other two sides. Suppose that the excircle tangent to \overline{B C} is internally tangent to \omega, and the other two excircles are both externally tangent to \omega. Find the minimum possible value of the perimeter of \triangle A B C.