
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:

\[
\left\{
\begin{array}{l}
x^{3}+y^{2}-3y+1<0 \\
3x^{3}-y^{2}+3y>0
\end{array}
\right.
\]

Provide the product of all the \(y\) values for all such pairs.

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