
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  $(O)$  denote a circle with a chord  $AB$ , and let  $W$  be the midpoint of the minor arc  $AB$ . Let  $C$  stand for an arbitrary point on the major arc  $AB$ . The tangent to the circle  $(O)$  at  $C$  meets the tangents at  $A$  and  $B$  at points  $X$  and  $Y$ , respectively. The lines  $W X$  and  $W Y$  meet  $AB$  at points  $N$  and  $M$  , respectively. Does the length of segment  $NM$  depend on position of  $C$  ?

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