
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
Starting from a given positive integer \( n_{0} \), two players (A and B) take turns selecting integers \( n_{1}, n_{2}, \cdots \) according to the following rules:

1. When \( n_{2k} \) is selected, player A can choose any integer \( n_{2k+1} \) such that \( n_{2k} \leq n_{2k+1} \leq n_{2k}^{2} \).
2. When \( n_{2k+1} \) is selected, player B can choose \( n_{2k+2} \) such that \( \frac{n_{2k+1}}{n_{2k+2}} \) is exactly a positive integer power of a prime.

The game is won by A if they pick 1990, and B wins if they pick 1 first. 

Question:
(1) For which values of \( n_{0} \) does A have a winning strategy?
(2) For which values of \( n_{0} \) does B have a winning strategy?
(3) For which values of \( n_{0} \) do neither player have a guaranteed winning strategy?

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