
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 $a$ for which the following system has a real solution $(x,y)$:
\begin{align*}
\sqrt{x^2y^2} &= a^a, \\
\log_a (x^{\log_a y}) + 2 \cdot \log_a (y^{\log_a x}) &= 6a^3.
\end{align*}

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