
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
 $a, b, c$  are positive real numbers such that   $$ (\sqrt {ab}-1)(\sqrt {bc}-1)(\sqrt {ca}-1)=1 $$  At most, how many of the numbers:   $$ a-\frac {b}{c}, a-\frac {c}{b}, b-\frac {a}{c}, b-\frac {c}{a}, c-\frac {a}{b}, c-\frac {b}{a} $$   can be bigger than  $1$ ?

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