
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
It is easy to see that there is no triplet of numbers \( x, y, z \) that satisfy the following system of equations. Determine the values of \( x, y, z \) such that the sum of the squares of the differences between the two sides of each equation is minimized:

$$
\begin{gathered}
x+y+z=10 \\
x+y-z=7 \\
x-y+z=6 \\
-x+y+z=5
\end{gathered}
$$

Present the answer in LaTex format: \boxed{Your answer}
user