
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
Determine all quadruplets $(a, b, c, d)$ of strictly positive real numbers satisfying $a b c d=1$ and:

\[ 
a^{2012} + 2012b = 2012c + d^{2012} 
\]
and 
\[ 
2012a + b^{2012} = c^{2012} + 2012d 
\]

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