
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 \( v(X) \) be the sum of elements of a nonempty finite set \( X \), where \( X \) is a set of numbers. Calculate the sum of all numbers \( v(X) \) where \( X \) ranges over all nonempty subsets of the set \(\{1,2,3, \ldots, 16\}\).

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