
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
For every permutation \(a_1, a_2, \ldots, a_n\) of \(1, 2, 4, 8, \ldots, 2^{n-1}\), form the product of all \(n\) partial sums \(a_1 + a_2 + \ldots + a_k\). Find the sum of the inverses of all these products.

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