7. Let $t=\left(\frac{1}{2}\right)^{x}+\left(\frac{2}{3}\right)^{x}+\left(\frac{5}{6}\right)^{x}$, then the sum of all real solutions of the equation $(t-1)(t-2)(t-3)=0$ with respect to $x$ is $\qquad$ .
7. 4 Detailed Explanation: Let $f(x)=\left(\frac{1}{2}\right)^{x}+\left(\frac{2}{3}\right)^{x}+\left(\frac{5}{6}\right)^{x}$, which can be rewritten as $f(x)=\left(\frac{3}{6}\right)^{x}+\left(\frac{4}{6}\right)^{x}+\left(\frac{5}{6}\right)^{x}$. It can be observed that the function $f(x)$ is a decreasing function on $\mathbf{R}$, with $f(3)=1$, $f(1)=2$, and $f(0)=3$. The solutions to the equation $(t-1)(t-2)(t-3)=0$ with respect to $x$ are $0, 1, 3$.