
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
Let \( x \), \( y \), and \( z \) be real numbers that satisfy the following system:
\[
\begin{aligned}
x + 3y + 6z &= 1 \\
xy + 2xz + 6yz &= -8 \\
xyz &= 2
\end{aligned}
\]

Find the smallest possible value of \( x + y + z \).

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