
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
Convert from kilometers to miles. In the problem 3.125, the Fibonacci numeral system was introduced as being useful when converting distances from kilometers to miles or vice versa.

Suppose we want to find out how many miles are in 30 kilometers. For this, we represent the number 30 in the Fibonacci numeral system:

$$
30=21+8+1=F_{8}+F_{6}+F_{2}=(1010001)_{\mathrm{F}}
$$

Now we need to shift each number one position to the right, obtaining

$$
F_{7}+F_{5}+F_{1}=13+5+1=19=(101001)_{\mathrm{F}}
$$

So, the estimated result is 19 miles. (The correct result is approximately 18.46 miles.) Similarly, conversions from miles to kilometers are done.

Explain why this algorithm works. Verify that it gives a rounded number of miles in $n$ kilometers for all $n \leqslant 100$, differing from the correct answer by less than $2 / 3$ miles.

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