
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
Find the set of all real number pairs \((x, y)\) that satisfy the following system of equations and inequalities:
\[
\begin{cases}
4^{-x} + 27^{-y} = \frac{5}{6} \\
\log_{27} y - \log_{4} x \geq \frac{1}{6} \\
27^y - 4^x \leq 1
\end{cases}
\]

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