
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
The given problem states: "Given the solution set of the inequality $ax^2 - bx + c > 0$ with respect to $x$ is $(1, 2)$, solve the inequality $cx^2 - bx + a > 0$ with respect to $x$." The provided solution is as follows: From $ax^2 - bx + c > 0$, we obtain $a - b(\frac{1}{x}) + c(\frac{1}{x})^2 > 0$. Let $t = \frac{1}{x}$, then $t \in (\frac{1}{2}, 1)$, thus the solution set of the inequality $cx^2 - bx + a > 0$ is $(\frac{1}{2}, 1)$. By analogy with the given solution, find the solution set of the inequality $\frac{k}{x+a} + \frac{x+b}{x+c} < 0$ with respect to $x$ if the solution set of $\frac{kx}{ax-1} + \frac{bx-1}{cx-1} < 0$ is $(-2, -1) \cup (2, 3)$.

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