
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

In a convex quadrilateral \(ABCD\), a diagonal \(BD\) is drawn, and circles are inscribed in each of the resulting triangles \(ABD\) and \(BCD\). A line passing through vertex \(B\) and the center of one of the circles intersects side \(DA\) at point \(M\). It is given that \(AM=\frac{25}{7}\) and \(MD=\frac{10}{7}\). Similarly, a line passing through vertex \(D\) and the center of the other circle intersects side \(BC\) at point \(N\). It is given that \(BN=\frac{3}{2}\) and \(NC=\frac{9}{2}\).

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

b) Find the lengths of sides \(AB\) and \(CD\) if it is additionally known that the given circles touch each other.

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