
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
For a certain value of $k$, the system
\[
\begin{align*}
x + ky - z &= 0, \\
4x + 2ky + 3z &= 0, \\
3x + 6y + 2z &= 0
\end{align*}
\]
has a solution where $x$, $y$, and $z$ are all nonzero. Find $\frac{xz}{y^2}.$

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