
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
Given a convex quadrilateral \(ABCD\). Let \(P\) be the center of the circle inscribed in triangle \(ABD\), and \(Q\) be the center of the circle inscribed in triangle \(CBD\). Ray \(BP\) intersects side \(DA\) at point \(M\), and ray \(DQ\) intersects side \(BC\) at point \(N\). It is given that \(AM = \frac{8}{3}\), \(DM = \frac{4}{3}\), \(BN = \frac{6}{5}\), and \(CN = \frac{9}{5}\).

a) Find the ratio \(AB : CD\).

b) Suppose additionally that the given circles are tangent to each other. Find the lengths of sides \(AB\) and \(CD\).

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