
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
Alice and Bob play the following "point guessing game." First, Alice marks an equilateral triangle \(ABC\) and a point \(D\) on segment \(BC\) satisfying \(BD = 3\) and \(CD = 5\). Then, Alice chooses a point \(P\) on line \(AD\) and challenges Bob to mark a point \(Q \neq P\) on line \(AD\) such that \(\frac{BQ}{QC} = \frac{BP}{PC}\). Alice wins if and only if Bob is unable to choose such a point. If Alice wins, what are the possible values of \(\frac{BP}{PC}\) for the \(P\) she chose?

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