
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
An equilateral triangle is tiled with  $n^2$  smaller congruent equilateral triangles such that there are  $n$  smaller triangles along each of the sides of the original triangle.  For each of the small equilateral triangles, we randomly choose a vertex  $V$  of the triangle and draw an arc with that vertex as center connecting the midpoints of the two sides of the small triangle with  $V$  as an endpoint. Find, with proof, the expected value of the number of full circles formed, in terms of  $n.$ ![Image](http://s3.amazonaws.com/classroom.artofproblemsolving.com/Images/Transcripts/497b4e1ef5043a84b433a5c52c4be3ae.png)

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