
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 with bases respectively equal to \(a\) and \(b\) (\(a > b\)), a line parallel to the bases is drawn through the intersection point of the diagonals. Find the length of the segment of this line intercepted by the non-parallel sides of the trapezoid.

Given (Fig. 33): \(ABCD\) is a trapezoid, \(|AD| = a\), \(|BC| = b\), \([AC]\) and \([BD]\) are diagonals, \((PQ) \parallel (AD)\), \(M \in (PQ)\), \(M = [AC] \cap [BD]\), \(P \in [AB]\), \(Q \in [CD]\).

Find: \(|PQ|\).

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