
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  $ABCD$  be an isosceles trapezoid with bases  $AB=5$  and  $CD=7$  and legs  $BC=AD=2 \sqrt{10}.$  A circle  $\omega$  with center  $O$  passes through  $A,B,C,$  and  $D.$  Let  $M$  be the midpoint of segment  $CD,$  and ray  $AM$  meet  $\omega$  again at  $E.$  Let  $N$  be the midpoint of  $BE$  and  $P$  be the intersection of  $BE$  with  $CD.$  Let  $Q$  be the intersection of ray  $ON$  with ray  $DC.$  There is a point  $R$  on the circumcircle of  $PNQ$  such that  $\angle PRC = 45^\circ.$  The length of  $DR$  can be expressed in the form  $\frac{m}{n}$  where  $m$  and  $n$  are relatively prime positive integers. What is  $m+n$ ?

*Author: Ray Li*

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