
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  $v$  and  $w$  be distinct, randomly chosen roots of the equation  $z^{1997}-1=0.$   Let  $m/n$  be the probability that  $\sqrt{2+\sqrt{3}}\le |v+w|,$  where  $m$  and  $n$  are relatively prime positive integers.  Find  $m+n.$ 

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