
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 all pairs of integers \((x, y)\) that satisfy the system of inequalities

\[
\begin{cases}
x^3 + y^2 - 3y + 1 < 0 \\
3x^3 - y^2 + 3y > 0
\end{cases}
\]

In the answer, specify the product of all \(y\) for all such pairs.

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