
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
On the faces \(BCD, ACD, ABD,\) and \(ABC\) of a tetrahedron \(ABCD\), points \(A_1, B_1, C_1,\) and \(D_1\) are marked, respectively. It is known that the lines \(AA_1, BB_1, CC_1,\) and \(DD_1\) intersect at point \(P\) with \(\frac{AP}{A_1P} = \frac{BP}{B_1P} = \frac{CP}{C_1P} = \frac{DP}{D_1P} = r\). Find all possible values of \(r\).

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