
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
The diagonals \( AC \) and \( BD \) of an isosceles trapezoid \( ABCD \) intersect at point \( O \). It is known that \( AD:BC = 3:2 \). A circle \(\omega\) with center \( O \), passing through the vertices \( A \) and \( D \), intersects the extension of the base \( BC \) beyond point \( B \) at point \( K \). It turns out that \( BK = BO \). Find the ratio of the base \( AD \) to the radius of the circle \(\omega\).

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