
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
Two players \(A\) and \(B\) play a game in which they choose numbers alternately according to the following rule: At the beginning, an initial natural number \(n_{0}>1\) is given. Knowing \(n_{2k}\), player \(A\) may choose any \(n_{2k+1} \in \mathbb{N}\) such that
\[ n_{2k} \leq n_{2k+1} \leq n_{2k}^{2}. \]
Then player \(B\) chooses a number \(n_{2k+2} \in \mathbb{N}\) such that
\[ \frac{n_{2k+1}}{n_{2k+2}}=p^{r}, \]
where \(p\) is a prime number and \(r \in \mathbb{N}\). 

It is stipulated that player \(A\) wins the game if he (she) succeeds in choosing the number 1990, and player \(B\) wins if he (she) succeeds in choosing 1.

For which natural numbers \(n_{0}\) can player \(A\) manage to win the game, for which \(n_{0}\) can player \(B\) manage to win, and for which \(n_{0}\) can players \(A\) and \(B\) each force a tie?

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