
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 semicircle with a diameter \(AB = 2r\) and center \(O\) has a tangent at point \(M\). This tangent intersects the tangents drawn from points \(A\) and \(B\) at points \(A'\) and \(B'\), respectively. The perpendicular projection of \(M\) onto the diameter \(AB\) is \(P\). Let \(x=\frac{AP}{PB} (>0)\).

Rotate the trapezoid \(AA'B'B\) and the triangle \(A'OB'\) around \(AB\) to form solids of revolution. Denote the volumes of these solids by \(V_1\) and \(V_2\), respectively. Express these volumes as functions of \(r\) and \(x\). Determine \(x\) such that \(\frac{V_2}{V_1}=m\), where \(m\) is a given positive number.

What is the maximum value \(m\) can take, and what is the corresponding position of \(M\)?

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