
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 function  $f$  is defined on the complex numbers by  $f(z)=(a+bi)z,$  where  $a$  and  $b$  are positive numbers.  This function has the property that the image of each point in the complex plane is equidistant from that point and the origin.  Given that  $|a+bi|=8$  and that  $b^2=m/n,$  where  $m$  and  $n$  are relatively prime positive integers.  Find  $m+n.$ 

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