Example 4 Given real numbers $x_{1}, x_{2}, x_{3}$ satisfy $x_{1}+\frac{1}{2} x_{2}+$ $\frac{1}{3} x_{3}=1$ and $x_{1}^{2}+\frac{1}{2} x_{2}^{2}+\frac{1}{3} x_{3}^{2}=3$, then the minimum value of $x_{3}$ is $\qquad$ . (Example 7 in [1])
By Cauchy-Schwarz inequality, we have $\left(1+\frac{1}{2}\right)\left(x_{1}^{2}+\frac{1}{2} x_{2}^{2}\right) \geqslant$ $\left(x_{1}+\frac{1}{2} x_{2}\right)^{2}$, i.e., $\frac{3}{2}\left(3-\frac{1}{3} x_{3}^{2}\right) \geqslant\left(1-\frac{1}{3} x_{3}\right)^{2}, \therefore$ $-\frac{21}{11} \leqslant x_{3} \leqslant 3$, hence the minimum value of $x_{3}$ is $-\frac{21}{11}$. (At this point, $x_{1}=$ $\left.x_{2}=\frac{12}{11}\right)$