
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
Heron's Formula states that if a triangle has side lengths \(a\), \(b\), and \(c\), then its area equals \(\sqrt{s(s-a)(s-b)(s-c)}\), where \(s=\frac{1}{2}(a+b+c)\) is called the semi-perimeter of the triangle.

(a) In the diagram, \(\triangle ABC\) has side lengths \(AB=20\), \(BC=99\), and \(AC=101\). If \(h\) is the perpendicular distance from \(A\) to \(BC\), determine the value of \(h\).

(b) In the diagram, trapezoid \(PQRS\) has \(PS\) parallel to \(QR\). Also, \(PQ=7\), \(QR=40\), \(RS=15\), and \(PS=20\). If \(x\) is the distance between parallel sides \(PS\) and \(QR\), determine the value of \(x\).

(c) The triangle with side lengths 3, 4, and 5 has the following five properties:
- its side lengths are integers,
- the lengths of its two shortest sides differ by one,
- the length of its longest side and the semi-perimeter differ by one,
- its area is an integer, and
- its perimeter is less than 200.

Determine all triangles that have these five properties.

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