
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 \( f(x, y) = x^2 + 2x + y^2 + 4y \). Let \( (x_1, y_1), (x_2, y_2), (x_3, y_3), \) and \( (x_4, y_4) \) be the vertices of a square with side length one and sides parallel to the coordinate axes. What is the minimum value of \( f(x_1, y_1) + f(x_2, y_2) + f(x_3, y_3) + f(x_4, y_4) \)?

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