
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  $n$  be a positive integer. A frog starts on the number line at  $0$ . Suppose it makes a finite sequence of hops, subject to two conditions:  	
- The frog visits only points in  $\{1, 2, \dots, 2^n-1\}$ , 	each at most once.
- The length of each hop is in  $\{2^0, 2^1, 2^2, \dots\}$ . 	(The hops may be 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  $S$ ?

*Ashwin Sah*

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