
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 that \( 15 * \star k \) is a given integer and the equation \( x^{2} - 2y^{2} = k \) has integer solutions \( x, y \), find the integer solutions \( t, u \) of the equation \( t^{2} - 2u^{2} = -k \).

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