
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 \(\mathcal{C}_{1}\) and \(\mathcal{C}_{2}\) be concentric circles, with \(\mathcal{C}_{2}\) in the interior of \(\mathcal{C}_{1}\). From a point \(A\) on \(\mathcal{C}_{1}\) one draws the tangent \(AB\) to \(\mathcal{C}_{2}\) \((B \in \mathcal{C}_{2})\). Let \(C\) be the second point of intersection of ray \(AB\) and \(\mathcal{C}_{1}\), and let \(D\) be the midpoint of \(\overline{AB}\). A line passing through \(A\) intersects \(\mathcal{C}_{2}\) at \(E\) and \(F\) in such a way that the perpendicular bisectors of \(\overline{DE}\) and \(\overline{CF}\) intersect at a point \(M\) on line \(AB\). Find, with proof, the ratio \(AM / MC\).

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