HMMT 2025 A7

There exists a unique triple $(a,b,c)$ of positive real numbers that satisfies the equations

$$2(a^{2} + 1) = 3(b^{2} + 1) = 4(c^{2} + 1) \mathrm{and} ab + bc + ca = 1.$$

Compute $a + b + c$ .