
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 two concentric circles, with \(\mathcal{C}_2\) inside \(\mathcal{C}_1\). Let \(A\) be a point on \(\mathcal{C}_1\) and \(B\) be a point on \(\mathcal{C}_2\) such that the line segment \(AB\) is tangent to \(\mathcal{C}_2\). Let \(C\) be the second point of intersection of the line segment \(AB\) with \(\mathcal{C}_1\), and let \(D\) be the midpoint of \(AB\). A line passing through \(A\) intersects \(\mathcal{C}_2\) at \(E\) and \(F\) such that the perpendicular bisectors of segments \(DE\) and \(CF\) intersect at a point \(M\) on the line segment \(AB\). Determine the ratio \(AM / MC\).

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