
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  $K{}$  be an equilateral triangle of unit area, and choose  $n{}$  independent random points uniformly from  $K{}$ . Let  $K_n$  be the intersection of all translations of  $K{}$  that contain all the selected points. Determine the expected value of the area of  $K_n.$ 

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