
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
Inside a tetrahedron \( A B C D \) there is a point \( O \) such that the lines \( A O, B O, C O, D O \) intersect the faces \( B C D, A C D, A B D, A B C \) of the tetrahedron at points \( A_{1}, B_{1}, C_{1}, D_{1} \) respectively, and the ratios

\[ 
\frac{A O}{A_{1} O}, \frac{B O}{B_{1} O}, \frac{C O}{C_{1} O}, \frac{D O}{D_{1} O} 
\]

are all equal to the same number. Find all the possible values that this number can take.

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