
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
Currently, the exchange rates for the dollar and euro are as follows: $D = 6$ yuan and $E = 7$ yuan. The People's Bank of China determines these exchange rates regardless of market conditions and follows a strategy of approximate currency parity. One bank employee proposed the following scheme for changing the exchange rates. Each year, the exchange rates can be adjusted according to the following four rules: Either change $D$ and $E$ to the pair $(D + E, 2D \pm 1)$, or to the pair $(D + E, 2E \pm 1)$. Moreover, it is prohibited for the dollar and euro rates to be equal at the same time.

For example: From the pair $(6, 7)$, after one year the following pairs are possible: $(13, 11)$, $(11, 13)$, $(13, 15)$, or $(15, 13)$. What is the smallest possible value of the difference between the higher and lower of the simultaneously resulting exchange rates after 101 years?

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