
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
Find all natural number which can be expressed in
 $ \frac{a\plus{}b}{c}\plus{}\frac{b\plus{}c}{a}\plus{}\frac{c\plus{}a}{b}$ 
where  $ a,b,c\in \mathbb{N}$  satisfy
 $ \gcd(a,b)\equal{}\gcd(b,c)\equal{}\gcd(c,a)\equal{}1$ 

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