
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  $d$  be a natural number. Given two natural numbers  $M$  and  $N$  with  $d$  digits,  $M$  is a friend of  $N$  if and only if the  $d$  numbers obtained substituting each one of the digits of  $M$  by the digit of  $N$  which is on the same position are all multiples of  $7$ . Find all the values of  $d$  for which the following condition is valid:
For any two numbers  $M$  and  $N$  with  $d$  digits,  $M$  is a friend of  $N$  if and only if  $N$  is a friend of  $M$ .

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