
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  $ X$ ,  $ Y$ ,  $ Z$  be distinct positive integers having exactly two digits in such a way that:

 $ X\equal{}10a \plus{}b$ 
 $ Y\equal{}10b \plus{}c$ 
 $ Z\equal{}10c \plus{}a$ 

( $ a,b,c$  are digits)

Find all posible values of  $ gcd(X,Y,Z)$ 

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