
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
Compute all values of $b$ such that the following system has a solution $(x,y)$ in real numbers:
\begin{align*}
\sqrt{xy} &= b^{2b}, \\
\log_b (x^{\log_b y}) + \log_b (y^{\log_b x}) &= 5b^5.
\end{align*}

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