
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
Jerry's favorite number is  $97$ .  He knows all kinds of interesting facts about  $97$ :


- $97$  is the largest two-digit prime.
- Reversing the order of its digits results in another prime.
- There is only one way in which  $97$  can be written as a difference of two perfect squares.
- There is only one way in which  $97$  can be written as a sum of two perfect squares.
- $\tfrac1{97}$  has exactly  $96$  digits in the [smallest] repeating block of its decimal expansion.
- Jerry blames the sock gnomes for the theft of exactly  $97$  of his socks.


A repunit is a natural number whose digits are all  $1$ .  For instance, \begin{align*}&1,&11,&111,&1111,&\vdots\end{align*} are the four smallest repunits.  How many digits are there in the smallest repunit that is divisible by  $97?$ 

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