
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
There are two positive integers \( A \) and \( B \) on a blackboard (\( A \geqslant B \)). Two players take turns replacing the integer on the blackboard. A player chooses a positive integer \( k \) such that \( A - kB \geqslant 0 \), removes \( A \) from the blackboard, and writes \( A - kB \) on the blackboard. The player who writes 0 on the blackboard wins. Determine all initial ratios of the two integers such that the first player always wins regardless of how the second player chooses.

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