AMO-Bench 2025 Problem 22

Let variables $x, y, c$ satisfy $x+y+c > \sqrt{13}$ with $x, y \ge 0$. Find the minimum value of $$\frac{(2+x^2)\,(4+y^2)\,\big(49+6c^2+4c\sqrt{3+3c^2}\big)} {\big(7\sqrt{3} c+14\sqrt{1+c^2}\big)\,(x+y)}.$$