
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 particular value of $k$, the system of equations
\begin{align*}
x + 2ky + 5z &= 0, \\
2x + ky + 4z &= 0, \\
3x + 6y + kz &= 0
\end{align*}
has a solution where $x$, $y$, and $z$ are all nonzero. Find the ratio $\frac{yz}{x^2}$.

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