
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
An isosceles trapezoid with bases  $a$  and  $c$  and altitude  $h$  is given.

a) On the axis of symmetry of this trapezoid, find all points  $P$  such that both legs of the trapezoid subtend right angles at  $P$ ;

b) Calculate the distance of  $p$  from either base;

c) Determine under what conditions such points  $P$  actually exist. Discuss various cases that might arise.

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