
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
Let $x_1$ and $x_2$ be the two real roots of the equation $2x^2 - 4mx + 2m^2 + 3m - 2 = 0$. For what value of $m$ does $x_1^2 + x_2^2$ have the minimum value, and what is this minimum value?

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