
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 trapezoid  $ABCD$  whose parallel sides  $AB$  and  $CD$  are in ratio  $\frac{AB}{CD}=\frac32$ ,  the points  $ N$  and  $M$  are marked on the sides  $BC$  and  $AB$  respectively, in such a way that  $BN = 3NC$  and  $AM = 2MB$  and segments  $AN$  and  $DM$  are drawn that intersect at point  $P$ , find the ratio between the areas of triangle  $APM$  and trapezoid  $ABCD$ .
![Image](https://cdn.artofproblemsolving.com/attachments/7/8/21d59ca995d638dfcb76f9508e439fd93a5468.png)

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