
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 \( a \) and \( b \) be the roots of the quadratic equation \( k(x^2 - 6x) + x + 8 = 0 \). Find the values of \( k \) for which \( a \) and \( b \) satisfy
\[
\frac{a}{b} + \frac{b}{a} = \frac{3}{7}.
\]
Determine \[
\frac{k_1}{k_2} + \frac{k_2}{k_1}.
\]

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