
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
Suppose $a$ is inversely proportional to the product of $b$ and $c$. Let $a_1, a_2$ be two non-zero values of $a$ such that $\frac{a_1}{a_2} = \frac{3}{4}$. Let the corresponding values of $b$ and $c$ for $a_1$ be $b_1, c_1$, and for $a_2$ be $b_2, c_2$. If $b_1 = 2b_2$ and $c_1, c_2$ are non-zero, find the value of $\frac{c_1}{c_2}$.

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