
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
4. Let  $v$  and  $w$  be two randomly chosen roots of the equation  $z^{1997} -1 = 0$  (all roots are equiprobable). Find the probability that  $\sqrt{2+\sqrt{3}}\le |u+w|$ 

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