
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 an isosceles trapezoid \(ABCD\) (\(BC \parallel AD\)), a circle is tangent to the base \(BC\), the lateral sides \(AB\) and \(CD\), and passes through the point \(M\), the intersection of diagonals \(AC\) and \(BD\). Find the radius of the circle if \(AD : BC = 9 : 7\) and the area of the trapezoid \(S = 8\).

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