
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
We draw a line $XY$ through point $C$, the midpoint of line segment $AB=2a$, such that it forms a $60^\circ$ angle with line $CB$. Choose point $P$ on line $XY$ so that the ratio of the lateral surfaces of the cones formed by rotating $PA$ and $PB$ around $XY$ is equal to a given value $m$. For which values of $m$ is this possible?

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