
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
Given the numbers $1, 2, 2^2, \cdots, 2^{n-1}$, for any permutation $\sigma=(x_1, x_2, \cdots, x_n)$ of these numbers, define $S_1(\sigma)=x_1$, $S_2(\sigma)=x_1 + x_2$, $S_3(\sigma)=x_1 + x_2 + x_3$, $\cdots$, $S_n(\sigma)=x_1 + x_2 + \cdots + x_n$, and let $Q(\sigma)=S_1(\sigma) \cdot S_2(\sigma) \cdot \cdots \cdot S_n(\sigma)$. Find $\sum \frac{1}{Q(\sigma)}$ (the sum taken over all permutations).

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