
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
Consider a positive integer \( n \). A frog starts at 0 on the number line and makes a sequence of hops, with the following conditions:
- The frog only visits points in the set \(\{1, 2, \ldots, 2^n - 1\}\), each point visited at most once.
- Each hop has a length chosen from the set \(\{2^0, 2^1, 2^2, \ldots\}\) and can be in either direction, left or right.

Let \( S \) be the sum of the (positive) lengths of all hops in the sequence. What is the maximum possible value of \( S \)?

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