
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
Given an initial integer \( n_0 > 1 \), two players A and B choose integers \( n_1, n_2, n_3, \ldots \) alternately according to the following rules: 

- Knowing \( n_{2k} \), player A chooses any integer \( n_{2k+1} \) such that \( n_{2k} \leq n_{2k+1} \leq n_{2k}^2 \).
- Knowing \( n_{2k+1} \), player B chooses any integer \( n_{2k+2} \) such that \( \frac{n_{2k+1}}{n_{2k+2}} = p^r \) for some prime \( p \) and integer \( r \geq 1 \).

Player A wins the game by choosing the number 1990; player B wins by choosing the number 1.

For which \( n_0 \) does:
(a) Player A have a winning strategy?
(b) Player B have a winning strategy?
(c) Neither player have a winning strategy?

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