
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
A kite is a quadrilateral whose diagonals are perpendicular. Let kite \(ABCD\) be such that \(\angle B = \angle D = 90^{\circ}\). Let \(M\) and \(N\) be the points of tangency of the incircle of \(ABCD\) to \(AB\) and \(BC\) respectively. Let \(\omega\) be the circle centered at \(C\) and tangent to \(AB\) and \(AD\). Construct another kite \(AB'C'D'\) that is similar to \(ABCD\) and whose incircle is \(\omega\). Let \(N'\) be the point of tangency of \(B'C'\) to \(\omega\). If \(MN' \| AC\), then what is the ratio of \(AB : BC\)?

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