
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
Consider a rectangle  $ABCD$  with  $BC = 2 \cdot AB$ . Let  $\omega$  be the circle that touches the sides  $AB$ ,  $BC$ , and  $AD$ . A tangent drawn from point  $C$  to the circle  $\omega$  intersects the segment  $AD$  at point  $K$ . Determine the ratio  $\frac{AK}{KD}$ .

*Proposed by Giorgi Arabidze, Georgia*

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