
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 a trapezoid \(A B C D\) with right angles at \(A\) and \(D\). Let \(A B = a\), \(C D = b\), and \(A D = h\).

1) What relationship must exist between \(a\), \(b\), and \(h\) for the diagonals \(A C\) and \(B D\) to be perpendicular?
2) Let \(M\) be the midpoint of \([A D]\). What relationship must exist between \(a\), \(b\), and \(h\) for the triangle \(B M C\) to be a right triangle at \(M\)?

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