
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 the lines \(AO, BO, CO,\) and \(DO\) intersect the faces \(BCD, ACD, ABD,\) and \(ABC\) at the points \(A_{1}, B_{1}, C_{1},\) and \(D_{1}\) respectively. Given that \(\frac{AO}{OA_{1}}=\frac{BO}{OB_{1}}=\frac{CO}{OC_{1}}=\frac{DO}{OD_{1}}=k\), find the value of \(k\).

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