Understanding the Concept:
Sometimes the quantity asked in the question is already directly present in the system of equations.
Instead of solving the entire system unnecessarily, observe carefully whether the required expression already appears.
Step 1: Observe the first equation carefully.
The system is:
\[
x+y+z=6
\]
\[
x+2y+3z=14
\]
\[
2x+y+z=7
\]
The question asks for:
\[
x+y+z
\]
But the first equation directly gives:
\[
x+y+z=6
\]
Therefore,
\[
\boxed{6}
\]