
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 Cartesian coordinate plane $xOy$, there are two sets of points:
\[ A = \left\{ (x, y) \mid x^2 + y^2 = 2(x + y) \right\} \]
\[ B = \{ (x, y) \mid kx - y + k + 3 \geq 0 \} \]
If the intersection of $A$ and $B$ is a single-element set, what is the value of $k$?

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