
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
Vertices  $A, B, C$  of a equilateral triangle of side  $1$  are in the surface of a sphere with radius  $1$  and center  $O$ . Let  $D$  be the orthogonal projection of  $A$  on the plane  $\alpha$  determined by points  $B, C, O$ . Let  $N$  be one of the intersections of the line perpendicular to  $\alpha$  passing through  $O$  with the sphere. Find the angle  $\angle DNO$ .

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