
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
Investigate the question: "Given the inequality $ax^2 - bx + c > 0$, let $y = \frac{1}{x}$, then $y \in \left(\frac{1}{2}, 1\right)$; thus, the solution set for the inequality $cx^2 - bx + a > 0$ is $\left(\frac{1}{2}, 1\right)$." Analogously, for the inequality $\frac{k}{x+a} + \frac{x+b}{x+c} < 0$ with the solution set $\left(-2, -1\right) \cup \left(2, 3\right)$, determine the solution set for the inequality $\frac{kx}{ax-1} + \frac{bx-1}{cx-1} < 0$.

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