
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

Define the efficiency of a natural number \( n \) as the proportion of all natural numbers from 1 to \( n \) (inclusive) that share a common divisor with \( n \) greater than 1. For example, the efficiency of the number 6 is \( \frac{2}{3} \).

a) Does a number exist with efficiency greater than 80%? If so, find the smallest such number.

b) Is there a number whose efficiency is the highest possible (i.e., not less than that of any other number)? If so, find the smallest such number.

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