
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 automata print pairs of natural numbers on cards. The automata work as follows:

1. The first automaton, after reading a card $(a, b)$, outputs a new card $(a+1, b+1)$.
2. The second automaton, after reading a card $(a, b)$, outputs a new card $\left(\frac{a}{2}, \frac{b}{2}\right)$ (it only works if both $a$ and $b$ are even).
3. The third automaton reads two cards $(a, b)$ and $(b, c)$, and outputs a new card $(a, c)$. Additionally, the automata can return all cards they have read.

Given an initial card $(5, 19)$, determine if it is possible to obtain:

1. $(1, 50)$
2. $(1, 100)$
3. Given an initial card $(a, b)$ where $a < b$, what value of $n$ allows obtaining the card $(1, n)$?

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