
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 solutions of the following system of equations in the set of non-negative numbers:

$$
\begin{aligned}
x^{3}+y^{3}+3 x y & =1 \\
x^{2}-y^{2} & =1
\end{aligned}
$$

Do we obtain other solutions if we only assume that $x+y>0$?

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