
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
The minimum value of the three-variable function 
$$
\begin{array}{l}
f(x, y, z) \\
=x^{2}+2 y^{2}+3 z^{2}+2 x y+4 y z+2 z x- \\
\quad 6 x-10 y-12 z
\end{array}
$$
is $\qquad$

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