
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
Three machines print pairs of natural numbers on cards. The machines operate as follows: The first machine, upon reading a card $(a ; b)$, outputs a new card $(a+1 ; b+1)$. The second machine, upon reading a card $(a ; b)$, outputs a card $(a / 2 ; b / 2)$ (it only works when $a$ and $b$ are even). The third machine, using two cards $(a ; b)$ and $(b ; c)$, outputs a card $(a ; c)$. Additionally, the machines return all the cards they read.

Initially, there is one card with the pair $(5 ; 19)$. Can you, using the machines in any order, obtain the card:
a) $(1 ; 50)$ ?
b) $(1 ; 100)$ ?

c) Suppose there is initially one card $(a ; b)$, where $a < b$, and we want to get a card $(1 ; n)$. For which $n$ is this possible?

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