
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 the tetrahedron \(ABCD\), there is a point \(O\) such that line segments \(AO, BO, CO, DO\) intersect the faces \(BCD, ACD, ABD, ABC\) at points \(A_1, B_1, C_1, D_1\) respectively, with \(\frac{AO}{A_1O} = \frac{BO}{B_1O} = \frac{CO}{C_1O} = \frac{DO}{D_1O} = k\). Determine all possible values of \(k\). (1968 Bulgarian Competition Problem)

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