
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
In the Cartesian coordinate system, $i$ and $j$ are unit vectors in the same direction as the positive x-axis and y-axis, respectively. Three points A, B, and C in the plane satisfy $\overrightarrow{AB} = 4\overrightarrow{i} + 3\overrightarrow{j}$ and $\overrightarrow{AC} = k\overrightarrow{i} - \frac{1}{2}\overrightarrow{j}$. Determine the number of possible values of the real number $k$ when points A, B, and C form a right-angled triangle.

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