
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 pairs of real numbers \((x, y)\) that satisfy the conditions:

$$
\left\{
\begin{array}{l}
3^{-x} y^{4}-2 y^{2}+3^{x} \leq 0 \\
27^{x}+y^{4}-3^{x}-1=0
\end{array}
\right.
$$

Compute the values of the expression \(x_{k}^{3}+y_{k}^{3}\) for each solution \((x_{k}, y_{k})\) of the system and find the minimum among them. In the answer, specify the found minimum value, if necessary rounding it to two decimal places. If the original system has no solutions, write the digit 0 in the answer field.

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