
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 xOy, $\overrightarrow{i}$ and $\overrightarrow{j}$ are the unit vectors in the same direction as the positive x-axis and y-axis, respectively. In the right-angled triangle ABC, if $\overrightarrow{AB} = \overrightarrow{i} + k\overrightarrow{j}$ and $\overrightarrow{AC} = 2\overrightarrow{i} + \overrightarrow{j}$, and $\angle C = 90^\circ$, then the value of $k$ is \_\_\_\_\_\_.

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