
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
A circle $K$ is internally tangent to a smaller circle $k$ whose radius is half of $K$'s radius. Construct a chord in $K$ that is perpendicular to the line connecting the centers of the two circles, and is divided into three equal parts by its intersection points with the circle $k$.

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