
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
Let \(CD\) be a chord of a circle \(\Gamma_{1}\) and \(AB\) a diameter of \(\Gamma_{1}\) perpendicular to \(CD\) at \(N\) with \(AN > NB\). A circle \(\Gamma_{2}\) centered at \(C\) with radius \(CN\) intersects \(\Gamma_{1}\) at points \(P\) and \(Q\), and the segments \(PQ\) and \(CD\) intersect at \(M\). Given that the radii of \(\Gamma_{1}\) and \(\Gamma_{2}\) are 61 and 60 respectively, find the length of \(AM\).

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