
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
Consider the equation
$$
\sqrt{3 x^{2}-8 x+1}+\sqrt{9 x^{2}-24 x-8}=3.
$$

It is known that the largest root of the equation is $-k$ times the smallest root. Find $k$.

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