Triangle $\mathrm{△}ABC$ has side lengths $AB=120,BC=220$, and $AC=180$. Lines ${\ell }_A,{\ell }_B$, and ${\ell }_C$ are drawn parallel to $\overline{BC},\overline{AC}$, and $\overline{AB}$, respectively, such that the intersections of ${\ell }_A,{\ell }_B$, and ${\ell }_C$ with the interior of $\mathrm{△}ABC$ are segments of lengths $55,45$, and $15$, respectively. Find the perimeter of the triangle whose sides lie on lines ${\ell }_A,{\ell }_B$, and ${\ell }_C$.