
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
Consider functions  $f$  from the whole numbers (non-negative integers) to the whole numbers that have the following properties: $\bullet$  For all  $x$  and  $y$ ,  $f(xy) = f(x)f(y)$ , $\bullet$   $f(30) = 1$ , and $\bullet$  for any  $n$  whose last digit is  $7$ ,  $f(n) = 1$ .
Obviously, the function whose value at  $n$  is  $ 1$  for all  $n$  is one such function. Are there any others? If not, why not, and if so, what are they?

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