求和: $S=\sqrt{1+\frac{4}{1^2}+\frac{4}{3^2}}+\sqrt{1+\frac{4}{2^2}+\frac{4}{4^2}}+\sqrt{1+\frac{4}{3^2}+\frac{4}{5^2}}+\sqrt{1+\frac{4}{4^2}+\frac{4}{6^2}}+\cdots +\sqrt{1+\frac{4}{{10}^2}+\frac{4}{{12}^2}}$.