<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 In the Cartesian coordinate system, if both the x-coordinate and y-coordinate of a point are integers, then the point is called an integer point. If the intersection point of the graphs of the functions y = 2x - 1 and y = kx + k is an integer point, calculate the number of integer values that k can take. Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
4
In the Cartesian coordinate system, if both the x-coordinate and y-coordinate of a point are integers, then the point is called an integer point. If the intersection point of the graphs of the functions y = 2x - 1 and y = kx + k is an integer point, calculate the number of integer values that k can take.
test