
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 $C$ be the [graph](https://artofproblemsolving.com/wiki/index.php/Graph) of $xy = 1$, and denote by $C^*$ the [reflection](https://artofproblemsolving.com/wiki/index.php/Reflection) of $C$ in the line $y = 2x$.  Let the [equation](https://artofproblemsolving.com/wiki/index.php/Equation) of $C^*$ be written in the form
\[12x^2 + bxy + cy^2 + d = 0.\]
Find the product $bc$.

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