
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. In JMO kingdom there are  $2^n$  citizens and a king. In terms of currency, the kingdom uses paper bills with value \  $$ 2^n $ and coins with value \$  $ 2^a(a=0,1\ldots ,n-1)$ . Every citizen has infinitely many paper bills. Let the total number of coins in the kingdom be  $S$ . One fine day, the king decided to implement a policy which is to be carried out every night:

- Each citizen must decide on a finite amount of money based on the coins that he currently has, and he must pass that amount to either another citizen or the king;
- Each citizen must pass exactly \ $1 more than the amount he received from other citizens.


Find the minimum value of $ S$ such that the king will be able to collect money every night eternally.

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