
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 isosceles trapezoid ABCD (AB parallel to DC, and BC = AD), let AB = \(a\), CD = \(c\) and let the perpendicular distance from A to CD be \(h\). Show how to construct all points \(X\) on the axis of symmetry such that \(\angle BXC = \angle AXD = 90^\circ\). Find the distance of each such \(X\) from AB and from CD. What is the condition for such points to exist?

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