
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  $n$  be a positive integer.  In  $n$ -dimensional space, consider the  $2^n$  points whose coordinates are all  $\pm 1$ .  Imagine placing an  $n$ -dimensional ball of radius 1 centered at each of these  $2^n$  points.  Let  $B_n$  be the largest  $n$ -dimensional ball centered at the origin that does not intersect the interior of any of the original  $2^n$  balls.  What is the smallest value of  $n$  such that  $B_n$  contains a point with a coordinate greater than 2?

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