
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
If $x$, $y$, and $k$ are positive real numbers such that
\[
5 = k^2 \left(\dfrac{x^2}{y^2} + \dfrac{y^2}{x^2}\right) + 2k\left(\dfrac{x}{y} + \dfrac{y}{x}\right),
\]
find the maximum possible value of $k$.

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