
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 \( a, b, c, \) and \( d \) be four distinct positive real numbers that satisfy the equations:

\[
\left(a^{2012}-c^{2012}\right)\left(a^{2012}-d^{2012}\right)=2011
\]

and

\[
\left(b^{2012}-c^{2012}\right)\left(b^{2012}-d^{2012}\right)=2011
\]

Find the value of \( (cd)^{2012} - (ab)^{2012} \).

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