
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
Given that \(ABCD\) is a tetrahedron, and points \(P, Q, R, S\) are on the edges \(AB, BD, CD, CA\) respectively, such that \(\frac{PA}{PB} = \frac{AS}{SC} = 1\) and \(\frac{BQ}{QD} = \frac{CR}{RD} = \frac{1}{2}\), the plane \(PQRS\) divides the tetrahedron into two parts with volumes \(V_1\) and \(V_2\) (\(V_1 < V_2\)). Find \(\frac{V_1}{V_2}\).

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