
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
For which integer pairs \((x, y)\) does the following system of inequalities hold?

$$
\begin{gathered}
y + \frac{1}{2} > \left| x^2 - 2x \right| \\
y \leq 2 - \left| x - 1 \right|
\end{gathered}
$$

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