
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
Given real numbers $x, y$ satisfying the conditions
$$
\begin{cases}
 1 \leq \lg(x y^2) \leq 2 \\
 -1 \leq \lg\left(\frac{x^{2}}{y}\right) \leq 2
\end{cases}
$$
find the maximum value of $\lg\left(\frac{x^{3}}{y^{4}}\right)$.

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