Solve by applying the simplex method to the dual problem.
Minimize
subject to
\[
\begin{array}{l}
C=10 x_{1}+7 x_{2}+12 x_{3} \\
x_{1}+x_{2}+2 x_{3} \geq 9 \\
2 x_{1}+x_{2}+x_{3} \geq 6 \\
x_{1}, x_{2}, x_{3} \geq 0
\end{array}
\]
Select the correct choice below and fill in any answer boxes within your choice.
$\operatorname{Min} C=\square$ at $x_{1}=\square, x_{2}=\square$, and $x_{3}=$
The optimal solution does not exist.

Step 1 ：The problem is a linear programming problem. The simplex method is a popular algorithm for solving such problems. However, the problem is given in the form of a minimization problem, and the constraints are in the form of inequalities that are greater than or equal to a certain value. The simplex method is typically used for maximization problems where the constraints are less than or equal to a certain value. Therefore, we need to convert this problem into a maximization problem and change the constraints accordingly. This is done by taking the dual of the problem.

Step 2 ：The dual of a linear programming problem is obtained by swapping the objective function with the right-hand side of the constraints, and swapping the coefficients of the variables in the objective function with the coefficients of the variables in the constraints. The inequalities are also reversed.

Step 3 ：After obtaining the dual problem, we can apply the simplex method to solve it. The solution to the dual problem will give us the solution to the original problem.

Step 4 ：The optimization failed and the problem appears to be unbounded. This means that there is no finite optimal solution to the problem. The simplex method is not able to find a solution because the feasible region is not bounded. This can happen when the constraints of the problem do not restrict the solution space enough to ensure a finite optimal solution.

Step 5 ：\(\boxed{\text{Final Answer: The optimal solution does not exist.}}\)