
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  $(v_1, ..., v_{2^n})$  be the vertices of an  $n$ -dimensional hypercube. Label each vertex  $v_i$  with a real number  $x_i$ . Label each edge of the hypercube with the product of labels of the two vertices it connects. Let  $S$  be the sum of the labels of all the edges. Over all possible labelings, find the minimum possible value of  $\frac{S}{x^2_1+ x^2_2+ ...+ x^2_n}$  in terms of  $ n$ .

Note: an  $n$  dimensional hypercube is a graph on  $2^n$  vertices labeled labeled with the binary strings of length  $n$ , where two vertices have an edge between them if and only if their labels differ in exactly one place. For instance, the vertices  $100$  and  $101$  on the  $3$  dimensional hypercube are connected, but the vertices  $100$  and  $111$  are not.

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