
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 bases \( BC \) and \( AD \) of the trapezoid \( ABCD \) are equal to \( a \) and \( b \). Four lines parallel to the bases are drawn. The first passes through the midpoints of the lateral sides, the second through the intersection point of the diagonals of the trapezoid, the third divides the trapezoid into two similar parts, and the fourth into two equal areas. Find the segments of these lines contained within the trapezoid and arrange the found values in ascending order.

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