
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
Given the equation 

\[ x_{1}^{2} + x_{2}^{2} = (x_{1} + x_{2})^{2} - 2 x_{1} x_{2} \]

and Vieta's formulas:

\[
\left\{
\begin{array}{l}
x_{1} + x_{2} = -(m+1) \\
x_{1} x_{2} = 2 m - 2
\end{array}
\right.
\]

where \( D = (m+3)^{2} \geq 0 \), find the value of \( x_{1}^{2} + x_{2}^{2} \). Then, determine \( y = (m-1)^{2} + 4 \) and find the minimum value of \( y \).

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