
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
For positive integers  $a$ ,  $b$ , and  $c$ , define \[ f(a,b,c)=\frac{abc}{\text{gcd}(a,b,c)\cdot\text{lcm}(a,b,c)}. \] We say that a positive integer  $n$  is  $f@$  if there exist pairwise distinct positive integers  $x,y,z\leq60$  that satisfy  $f(x,y,z)=n$ . How many  $f@$  integers are there?

*Proposed by Michael Ren*

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