
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
Points  $A$  and  $B$  are the endpoints of a diameter of a circle with center  $C$ . Points  $D$  and  $E$  lie on the same diameter so that  $C$  bisects segment  $\overline{DE}$ . Let  $F$  be a randomly chosen point within the circle. The probability that  $\triangle DEF$  has a perimeter less than the length of the diameter of the circle is  $\tfrac{17}{128}$ . There are relatively prime positive integers m and n so that the ratio of  $DE$  to  $AB$  is  $\tfrac{m}{n}.$  Find  $m + n$ .

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