
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  $\mathbb{N}_{\geqslant 1}$  be the set of positive integers. 
Find all functions  $f \colon \mathbb{N}_{\geqslant 1} \to \mathbb{N}_{\geqslant 1}$  such that, for all positive integers  $m$  and  $n$ :
\[\mathrm{GCD}\left(f(m),n\right) + \mathrm{LCM}\left(m,f(n)\right) =
\mathrm{GCD}\left(m,f(n)\right) + \mathrm{LCM}\left(f(m),n\right).\]

Note: if  $a$  and  $b$  are positive integers,  $\mathrm{GCD}(a,b)$  is the largest positive integer that divides both  $a$  and  $b$ , and  $\mathrm{LCM}(a,b)$  is the smallest positive integer that is a multiple of both  $a$  and  $b$ .

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