
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
Let \( k \) and \( t \) be constants. Consider the quadratic equation in \( x \):

\[ k x^{2}+(3-3 k) x+2 k-6=0 \]

that has only integer roots. Also consider the quadratic equation in \( y \):

\[ (k+3) y^{2}-15 y+t=0 \]

which has two positive integer roots \( y_{1} \) and \( y_{2} \).
(1) Find the value of \( k \);
(2) For which value of \( t \) does \( y_{1}^{2}+y_{2}^{2} \) have its minimum value?

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