
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
Given an isosceles trapezoid with bases $a$ and $b$, and height $h$:

(a) Find a point $P$ on the axis of symmetry of the trapezoid such that the angle subtended by the legs of the trapezoid at $P$ is a right angle.

(b) Calculate the distances from point $P$ to the two bases of the trapezoid.

(c) Discuss the conditions under which point $P$ can be constructed (i.e., the conditions under which point $P$ exists).

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