
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

Given \( x, y \in \mathbf{R} \), let the sets
$$
\begin{array}{l}
A=\left\{(x, y) \mid x^{2}-3xy+4y^{2}=\frac{7}{2}\right\}, \\
B=\{(x, y) \mid kx + y=2\} \text{ with } k>0.
\end{array}
$$

When \( A \cap B \) has only one element, what is \( k = \) ?

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