
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
\(ABCD\) is a square with side length 9. Let \(P\) be a point on \(AB\) such that \(AP: PB = 7:2\). Using \(C\) as the center and \(CB\) as the radius, a quarter circle is drawn inside the square. The tangent from \(P\) meets the circle at \(E\) and \(AD\) at \(Q\). The segments \(CE\) and \(DB\) meet at \(K\), while \(AK\) and \(PQ\) meet at \(M\). Find the length of \(AM\).

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