HMMT 2025 U30

Let $a$ , $b$ , and $c$ be real numbers satisfying the system of equations

$$a\sqrt{1 + b^{2}} +b\sqrt{1 + a^{2}} = \frac{3}{4},$$ $$b\sqrt{1 + c^{2}} +c\sqrt{1 + b^{2}} = \frac{5}{12},\mathrm{and}$$ $$c\sqrt{1 + a^{2}} +a\sqrt{1 + c^{2}} = \frac{21}{20}.$$

Compute $a$ .