
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 \(\Gamma_{1}\) and \(\Gamma_{2}\) be two concentric circles, with \(\Gamma_{2}\) inside \(\Gamma_{1}\). Let \(A\) and \(B\) be points on \(\Gamma_{1}\) and \(\Gamma_{2}\) respectively, such that \(AB\) is tangent to \(\Gamma_{2}\).

Let \(C\) be the second point of intersection of \(AB\) with \(\Gamma_{1}\) and let \(D\) be the midpoint of \(AB\). A line passing through \(A\) intersects \(\Gamma_{2}\) in \(E\) and \(F\) such that the perpendicular bisectors of \(DE\) and \(CF\) intersect at a point \(M\) on \(AB\). Calculate the ratio \(AM / MC\).

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