
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

The base of pyramid \( T ABCD \) is an isosceles trapezoid \( ABCD \) with the length of the shorter base \( BC \) equal to \( \sqrt{3} \). The ratio of the areas of the parts of the trapezoid \( ABCD \), divided by the median line, is \( 5:7 \). All the lateral faces of the pyramid \( T ABCD \) are inclined at an angle of \( 30^\circ \) with respect to the base. The plane \( AKN \), where points \( K \) and \( N \) are the midpoints of the edges \( TB \) and \( TC \) respectively, divides the pyramid into two parts. Find the volume of the larger part.

**(16 points)**

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