
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 
\[
5 = k^2 \left(\frac{x^2}{y^2} + \frac{y^2}{x^2}\right) + k \left(\frac{x}{y} + \frac{y}{x}\right),
\]
find the maximum possible value of \( k \).

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