
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 a trapezoid, the lengths of the parallel sides are \(a\) and \(b\), and the height is \(m\). What is the length of the line parallel to \(a\) and \(b\) that divides the trapezoid into two parts whose areas are in the ratio \(p: q\)?

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