What Can We Equate?

Geometry Level 5

The lengths of three sides of \(\triangle ABC\) are \(AB=a+3b\), \(BC=a+2b\), and \(CA=a+b\) where \(a\) is a positive integer and \(0<b\leq 1\). If the length of the altitude to side \(BC\) is \(a\), find the number of triangles, no pair being similar, which satisfy the conditions.

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