
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  $\mathbb{N}$  be the set of positive integers. Across all function  $f:\mathbb{N}\to\mathbb{N}$  such that   $$ mn+1\text{ divides } f(m)f(n)+1 $$   for all positive integers  $m$  and  $n$ , determine all possible values of  $f(101).$ 

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