
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
A triangular pyramid is cut by a plane into two polyhedra. Find the ratio of the volumes of these polyhedra if it is known that the cutting plane divides the edges converging at one vertex of the pyramid in the ratio $1:2, 1:2, 2:1$, counting from that vertex.

Given: $SABC$ is a pyramid, $S$ is the vertex, $P \in [SA]$, $Q \in [AB]$, and $R \in [AC]$, with $|AQ|:|QB| = 1:2$, $|AR|:|RC| = 1:2$, $|AP|:|PS| = 2:1$. $(PQR)$ is the cutting plane.

Find: $V_{1}:V_{2}$ - the ratio of the volumes of the resulting polyhedra.

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