If $f\left(x\right)=\{\begin{array}{ccc}\frac{6^x-3^x-2^x+1}{2{\sin }^2\left(\frac{x}{2}\right)},& \text{if}& x\ne 0\\ K,& \text{if}& x=0\end{array}$ is continuous at $x=0,$ then $K=$

$$\begin{gathered} K=f\left(0\right)=\underset{x\to 0}{Lt}f\left(x\right)=\underset{x\to 0}{Lt}\frac{(3^x-1)(2^x-1)}{2{\sin }^2\left(\frac{x}{2}\right)} \\ =\underset{x\to 0}{Lt}\frac{{\displaystyle \frac{(3^x-1)(2^x-1)}{x^2}}}{2{\displaystyle \frac{{\sin }^2\left(\frac{x}{2}\right)}{x^2}}}=\frac{\log 3.\log 2}{2.{\displaystyle \frac{1}{4}}}=2\log 3.\log 2 \end{gathered}$$