
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
\[
\begin{align*}
x + 2ky + 4z - w &= 0, \\
4x + ky + 2z + w &= 0, \\
3x + 5y - 3z + 2w &= 0, \\
2x + 3y + z - 4w &= 0
\end{align*}
\]
has a solution where $x, y, z, w$ are all nonzero. Find the value of $\frac{xy}{zw}$.

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