
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 that \( x \) and \( y \) are real numbers, find the minimum value of the function \( z = \sqrt{x^2 + y^2 - 2x - 2y + 2} + \sqrt{x^2 + y^2 - 4y + 4} \). Also, specify the real values of \( x \) and \( y \) when this minimum value is achieved.

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