
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
\[ x_{1}^{2} + x_{2}^{2} = (x_{1} + x_{2})^{2} - 2 x_{1} x_{2}, \]
but, according to Vieta's theorem
\[ \begin{cases}
  D \geq 0, \\
  x_{1} + x_{2} = -(m+1), \\
  x_{1} x_{2} = 2m - 2.
\end{cases} \]

Given that \( D = (m+3)^{2} \geq 0 \), we have
\[ x_{1}^{2} + x_{2}^{2} = (-(m+1))^{2} - 2(2m-2) = m^{2} + 2m + 1 - 4(m-1) = m^{2} - 2m + 5 = (m-1)^{2} + 4. \]

Hence, \( y = (m-1)^{2} + 4 \) and \( y_{\text{min}} = 4 \) when \( m = 1 \).

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