
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 with a given positive integer \( n_{0} > 1 \), players A and B take turns selecting numbers \( n_{1}, n_{2}, \cdots \) according to the following rules: 
- When \( n_{2k} \) is selected, player A can choose any integer \( n_{2k+1} \) such that \( n_{2k} \leqslant n_{2k+1} \leqslant n_{2k}^{2} \).
- When \( n_{2k+1} \) is selected, player B can choose \( n_{2k+2} \), where \( n_{2k+2} \) is an integer such that \( \frac{n_{2k+1}}{n_{2k+2}} \) is a positive integer power of a prime number.

Player A wins if they can first select 1990, and player B wins if they can first select 1. Determine:
1. For which \( n_{0} \) does player A have a winning strategy?
2. For which \( n_{0} \) does player B have a winning strategy?
3. For which \( n_{0} \) do both players lack a guaranteed winning strategy?

(From the 31st IMO)

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