
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
In the pyramid \(FABC\), \(AB = BC\), \(FB = FK\), where \(K\) is the midpoint of segment \(AC\), and the tangent of the angle between the planes \(FAB\) and \(ABC\) is related to the tangent of the angle between the planes \(FBC\) and \(ABC\) as \(1:3\). A plane \(\pi\), parallel to \(AB\), divides edge \(FC\) in the ratio \(1:4\) (counting from vertex \(F\)), and passes through the base \(O\) of the height \(FO\) of pyramid \(FABC\). Find the ratio of the volumes of the polyhedrons into which this plane divides the pyramid \(FABC\).

### Answer
\[ 5: 11 \text{ or } 1: 19 \]

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