
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
Triangle $\mathrm{XYZ}$ is inside square KLMN shown below so that its vertices each lie on three different sides of the square. It is known that:
- The area of square KLMN is 1.
- The vertices of the triangle divide three sides of the square into these ratios:
$$
\begin{array}{ll}
\mathrm{KX}: \mathrm{XL} & = 3 : 2 \\
\mathrm{KY}: \mathrm{YN} & = 4 : 1 \\
\mathrm{NZ}: \mathrm{ZM} & = 2 : 3
\end{array}
$$

What is the area of the triangle XYZ?

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