
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 \( C_1 \) and \( C_2 \) be two concentric circles, with \( C_2 \) inside \( C_1 \). Let \( A \) be a point on \( C_1 \) and \( B \) a point on \( C_2 \) such that the line segment \( AB \) is tangent to \( C_2 \). Let \( C \) be the second point of intersection of the line segment \( AB \) with \( C_1 \), and let \( D \) be the midpoint of \( AB \). A line passing through \( A \) intersects \( C_2 \) at points \( E \) and \( F \) such that the perpendicular bisectors of \( DE \) 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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