
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 \( k \) be a real number. In the plane Cartesian coordinate system, there are two sets of points \( A = \left\{ (x, y) \mid x^2 + y^2 = 2(x + y) \right\} \) and \( B = \left\{ (x, y) \mid kx - y + k + 3 \geq 0 \right\} \). If \( A \cap B \) is a unit set, determine the value of \( k \).

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