
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
Given three automates that deal with the cards with the pairs of natural numbers. The first, having got the card with ( $a,b)$ , produces new card with  $(a+1,b+1)$ , the second, having got the card with  $(a,b)$ , produces new card with  $(a/2,b/2)$ , if both  $a$  and  $b$  are even and nothing in the opposite case; the third, having got the pair of cards with  $(a,b)$  and  $(b,c)$  produces new card with  $(a,c)$ . All the automates return the initial cards also. Suppose there was  $(5,19)$  card initially. Is it possible to obtain 
a)  $(1,50)$ ? 
b)  $(1,100)$ ? 
c) Suppose there was  $(a,b)$  card initially  $(a<b)$ . We want to obtain  $(1,n)$  card. For what  $n$  is it possible?

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