
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
 $A,B,C$  are points in the plane such that  $\angle ABC=90^\circ$ . Circles with diameters  $BA$  and  $BC$  meet at  $D$ . If  $BA=20$  and  $BC=21$ , then the length of segment  $BD$  can be expressed in the form  $\frac{m}{n}$  where  $m$  and  $n$  are relatively prime positive integers. What is  $m+n$ ?

*Ray Li*

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