
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
Inside an angle of $30^{\circ}$ with vertex $A$, a point $K$ is chosen such that its distances to the sides of the angle are 1 and 2. Through point $K$, all possible lines intersecting the sides of the angle are drawn. Find the minimum area of a triangle that is cut off by one of these lines from the angle.

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