
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 values of \( c \) for which the following system has a solution \( (x, y) \) in real numbers:
\begin{align*}
\sqrt{x^2y^2} &= c^{2c}, \\
\log_c (x^{\log_c y}) + \log_c (y^{\log_c x}) &= 8c^4.
\end{align*}

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