
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 that $x, y \in (0, +\infty)$, $a \in \mathbb{R}$, if $x^3 + \ln{x} + 2a^2 = 0$ and $4y^3 + \ln{\sqrt{y}} + \ln{\sqrt{2}} + a^2 = 0$, then $\frac{x}{y} = \_\_\_\_\_\_.$

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